# Lorentz–like extension Mixing Higgs and Gauge Fields in a U(1) symmetric
model

Luis Alberto Wills-Toro1,2,3
https://orcid.org/0000-0001-8815-8782

1\. School of Mathematics, Universidad Nacional de Colombia,
Medellín, Colombia, e–mail<EMAIL_ADDRESS>
2\. Departamento de Física Teórica y del Cosmos, Universidad de Granada
3\. CIMAT

A program searching for symmetry structures behind some features of the
standard Model is launched. After discussing the program and addressing known
no-go theorems, we construct some concrete symmetry extensions involving both
gauge and Lorentz symmetries. We then provide a field theory model with such a
local symmetry, which is in turn invariant under local Lorentz
transformations. It recovers both gauge as well as some sort of Higgs-like
potential. We address the action of extended symmetries on the geometric
frame, and models with extra dimensions.
Keywords: Gauge Symmetry, Higher Spin Symmetry, Field Theories in Higher
Dimensions.

##  1 Introduction

We would like to launch a program to study novel symmetries in QFT models
aiming to bring light into diverse challenges (such as dark energy, dark
matter, neutrino masses, scale naturality, background independence, etc.) in
extending the Standard Model –SM– of particle physics:

We propose the following simple rules (already present in many endevors):

  * $\bullet$

We would like to bring naturality to rather ad hoc structures of the SM,
beyond the “internal $\oplus$ external” symmetry paradigm.

  * $\bullet$

We sponsor the effort of reducing the manifold particle brands: Gauge, Matter,
Higgs. It would be desirable if all particles (and fields, including those of
scalar and fermionic type) were in correspondence with specific symmetry
generators or leftovers of them in the low energy realm. We demand lagrangian
densities that are outcomes of invariants of the considered symmetries.

  * $\bullet$

We view some well established physical and mathematical structures (such as
minimal couplings, connections, Wick rotations, causality, spontaneous
symmetry breaking, inertial effects, etc.) might be just effective or emerging
outcomes of some (yet to uncover mathematical structures of) earlier stages of
the universe.

  * $\bullet$

The constraints of well known No-Go theorems should be carefully understood.
They allow for the consideration of symmetries that might break spontaneously,
and will not thereafter remain as S–Matrix symmetries in the forthcoming
descendants.

  * $\bullet$

We want to gain insight into possible relationships between space-time and
internal symmetries. But initially, we regard the space–time coordinates as
mere parametrizations of the 4D manifold with no obvious relationship to the
symmetries to be inserted through connections.

  * $\bullet$

We expect symmetries that are invariant under space–time dependent symmetry
transformations. At least initially, we have no guarantee that the symmetries
considered can be represented nontrivially through plain differential
operators in the space-time coordinates (internal symmetries and spinors are
reluctant to do so). Anyway, we might obtain concrete suggestions of possible
relations to theories in greater superspaces or with extra dimensions. In
fact, the use of higher spin fields, or the use of simple groups with
non–positive definite metric, or even the use of non-compact or non–simple
groups present serious challenges towards quantization. Only a careful study
under new settings, somehow beyond well known no–go theorems, will decide if
they lead to consistent QFT models.

  * $\bullet$

We expect that perhaps through spontaneous symmetry breaking or some other
mathematical structure we might land into the diverse structures of the
standard model and its coupling to classical gravity. In going down from more
restrictive symmetries we might expect to realize the emerging space-time
symmetries through differential operators in space-time, and to cast also an
emerging local invariance under reparametrizations in a 4D manifold, thereby
enriching the geometry of the manifold through the fields, and the dynamic of
the fields through the enriched geometry. Some of the symmetries, such as
Lorentz invariance are expected to survive only as global symmetry remnant.

  * $\bullet$

We would like to relate diverse particles into novel multiplets under the
considered novel symmetries. Since all particles of the SM are either in
fundamental (for matter and Higgs) or adjoint (for gauge fields)
representations of the Lorentz and internal symmetries, it seems that some
brand of supersymmetric extension might introduce the spinors in fundamental
representations.

It would be desirable that some mechanism is in place in order to guarantee
that besides the Electro–Weak (EW) gauge bosons, also the EW scalars are
massless when reaching the energy scales that trigger the EW symmetry breaking
[1], and that some structure affords in a consistent way the masses that
decouple further gauge fields in the low energy realm. The Higgs and the
electroweak gauge fields in the Standard Model (SM) have analogous
ultra–violet features. Furthermore, they entangle in a non–linear
recombination of degrees of freedom after a spontaneous symmetry breaking. We
contemplate as plausible that at higher energies some sort of symmetry might
have governed a mixing among the Higgs field $H_{\alpha},\bar{H^{\alpha}}$,
and the gauge field –$SU(2)$ connection– of isospin $A_{[i]\mu}$ (of
hypercharge 1, with $i$ adjoint $SU(2)$ representation index, $\mu$ space–time
index, with $g_{\mu\nu}\\!=\\!{\rm diag}(1,-1,-1,-1)\\!=\\!g^{\mu\nu}$), and
the gauge field –$U(1)$ connection– of Hypercharge symmetry $B_{\mu}$. These
fields have naive energy dimension 1, so we consider dimensionless generators
$T_{\alpha\mu},\hat{T}^{\,\,\alpha}_{\mu}$, such that:

$\displaystyle[T_{\alpha\mu},\bar{H}^{\beta}]\\!=\\!a^{\prime}\sigma[i]_{\alpha}^{\,\,\beta}A_{[i]\mu}\\!+\\!b^{\prime}\delta_{\alpha}^{\,\,\beta}B_{\mu}\\!+\\!\cdots,\
[T_{\alpha\mu},B_{\nu}]\\!=\\!c^{\prime}g_{\mu\nu}{H}_{\alpha}+\cdots,$ (1)
$\displaystyle[\hat{T}_{\mu}^{\
\alpha}\\!,{H}_{\beta}]\\!=\\!-\bar{a}^{\prime}\sigma[i]_{\beta}^{\,\,\alpha}A_{[i]\mu}\\!-\\!\bar{b}^{\prime}\delta^{\
\alpha}_{\beta}B_{\mu}\\!+\\!\cdots,\
[\hat{T}^{\,\,\alpha}_{\mu},B_{\nu}]=-\bar{c^{\prime}}g_{\mu\nu}\bar{H}^{\alpha}+\cdots,\
\ \ $ (2)
$\displaystyle[T_{\alpha\mu},A_{[i]\nu}]\\!=\\!f^{\prime}g_{\mu\nu}\sigma[i]_{\alpha}^{\,\,\beta}{H}_{\beta}\\!+\\!\cdots,\
[\hat{T}^{\,\,\alpha}_{\mu},A_{[i]\nu}]=-\bar{f}^{\prime}g_{\mu\nu}{\bar{H}}^{\beta}\sigma[i]_{\beta}^{\
\ \alpha}\\!+\\!\cdots.\ \ \ $ (3)

where $a^{\prime},b^{\prime},c^{\prime},f^{\prime}$ are constants.
Phenomenologically, we have hints of a non trivial relationship between
internal and external symmetries. For instance, we have particles (leptons)
whose diverse spin (external symmetry) representations respond differently
under electroweak interactions. Using representation theory we inquire about
the closure of the naive energy dimension 0 subalgebra including the
generators $T_{\alpha\mu},\hat{T}^{\,\,\alpha}_{\mu}$, which might include the
Lorentz subalgebra with generators $M_{\mu\nu}$ or a clone thereof, a
generator which is a real traceless symmetric tensor $T_{\mu\nu}$ (whose role
is not yet specified), a scalar generator $D$ (dilatations?), as well as the
isospin generators $T_{[i]}$, and hypercharge generator ${T_{[o]}}$:

$\displaystyle\,[T_{\alpha\mu},\hat{T}^{\,\,\beta}_{\nu}]$ $\displaystyle=$
$\displaystyle-{\,{\rm
i}\,}a\delta_{\alpha}^{\,\,\beta}M_{\mu\nu}+u\delta_{\alpha}^{\,\,\beta}T_{\mu\nu}$
(4) $\displaystyle-
qg_{\mu\nu}\delta_{\alpha}^{\,\,\beta}D-bg_{\mu\nu}\sigma[i]_{\alpha}^{\,\,\beta}T_{[i]}+(h/2)g_{\mu\nu}\delta_{\alpha}^{\,\,\beta}{T_{[o]}},$

where $a,u,q,b,h$ are constants. This resembles a sort of extension of
conformal symmetry. These type of extensions involving internal and external
symmetries have been generically examined under constraints set by:

Theorem [Coleman–Mandula No-Go][2] If G is a symmetry group of the S–Matrix of
a QFT model, where (i) For any $M>0$ there are only a finite number of
particle types with mass less than $M$; (ii) Scattering occurs at almost all
energies (except for perhaps some isolated set of energies); (iii) The
amplitudes for elastic two-body scattering are analytic functions of the
scattering angle at almost all energies and angles; then the generators of G
consist of only the generators of the Poincaré group P, and the scalar
generators of internal symmetries.

Nevertheless, in some instances, we can bypass some of the constraints set by
the Coleman–Mandula (CM) theorem [3]. Spontaneously broken symmetries of the
action: Once broken, such symmetries are not required to act on the S–Matrix
nor need to commute with it. Discrete symmetries are not ruled out either. In
massless theories, some arguments about antisymmetry in CM proof do not hold.
That is the case of, say, conformal symmetry [4][5]. The CM proof depends on
the use of just commutation relations. A wider algebraic frame can evade the
theorem. This is the case of $\mathbb{Z}_{2}$–graded or Lie superalgebras,
covered by the Haag–Łopuszański–Sohnius no–go theorem [6]. Analyticity of S by
scattering angle of two bodies in 1+1 dimensions is not satisfied, since we
have just forward and backward scattering.

Our inquiry include an initial stage of just massless fields, and a symmetry
enlarging the Lorentz, and then the Poincaré algebra. We expect also that some
spontaneous symmetry breaking might land in a model aligned with the
constraints set by the CM theorem or the corresponding generalization. We will
proceed letting the obtained outcomes hint us on how to extend familiar
structures. The main tools we promote are the Lie algebraic closure of the
proposed structures, their invariants, hypothetical covariant derivatives, the
associated curvature, and the construction of invariant lagrangian densities
associated with Lie algebraic invariants. We label and group sets of
generators associated with concrete (irreducible) representations under the
subgroups prominent in the low energy realm, and even introduce separate
labeling for their structure constants, so that we can trace the origin of
diverse contributions. In fact, the actual nature of the Lie algebraic
structures is initially unknown, so the question about simplicity or
semisimplicity and the quest for independent couplings for their simple
factors is still unsettled. In order to nurture and promote the natural
ingenuity of the approach, only at the end we explore relations and overlap
with diverse notorious advances in this and close realms.

One last thing needs to be discussed. The Higgs field
$H_{\alpha},\bar{H^{\alpha}}$ is in the fundamental representation of the
isospin symmetry. So, in order to have a Higgs field as a gauge field we need
to allow for a symmetry including generators in the fundamental
representations of $SU(2)$. The reader will find some parallelism to the
undertaking of supersymmetry, which involves the fundamental representations
of the Lorentz symmetry in the extended symmetry. Later stages of this program
will tackle this endeavor.

The transformation of fields in (1)-(4) involve both space–time and internal
symmetry groups. It might lead to a non trivial connection between internal
and external symmetries. We develope here this program first in the absence of
the $SU(2)$ symmetry to explore a simpler case, so we eliminate the indices
$\alpha,\beta,i,\cdots$. We will tackle in a forthcoming contribution the case
with $SU(2)$, then with two $SU(2)$, and extrapolate these results. Also, we
will include there extensions involving the fundamental representations of
$SU(2)$.

##  2 Extending the Dimensionless Subalgebras

The closure of an algebra including the Lorentz–like subagebra for
$M_{\mu\nu}$, the $U(1)$ symmetry with generator $T_{[o]}$, and the relation
(4) dropping the $SU(2)$ generator contributions is based on assuming the
following Ansätze: all generators indices are transformed as expected
according to their Lorentz and $U(1)$ indices. We assign only non trivial
$U(1)$ charge +1 to $T_{\mu}$, and -1 to $\hat{T}_{\nu}$. We do not establish
yet if $T_{\mu}$ and $\hat{T}_{\nu}$ build an adjoint pair. We assume that
Lorentz vectors are transformed into themselves under the action of
$T_{\mu\nu}$ (Lorentz vectors are invariant subspace under $T_{\mu\nu}$). This
leads, after renormalization, to the dimension 0 subalgebra, writing only the
nontrivial relations:

$\displaystyle\,[T_{\mu},\hat{T}_{\nu}]$ $\displaystyle=$
$\displaystyle\epsilon({\,{\rm
i}\,}M_{\nu\mu}+T_{\nu\mu}+(1/10)g_{\nu\mu}(4{T_{[o]}}+{\hat{T}_{[o]}}))+2\epsilon{T_{[o]}},$
(5) $\displaystyle\,[{T_{[o]}},T_{\rho}]$ $\displaystyle=$ $\displaystyle
T_{\rho},\ \ \ \ \ \ \ \ \ \ \ \,[{T_{[o]}},\bar{T}_{\rho}]=\
-\bar{T}_{\rho},$ (6) $\displaystyle\,[{\hat{T}_{[o]}},T_{\rho}]$
$\displaystyle=$ $\displaystyle T_{\rho},\ \ \ \ \ \ \ \ \ \ \
\,[{\hat{T}_{[o]}},\bar{T}_{\rho}]=\ -\bar{T}_{\rho},$ (7)
$\displaystyle\,[T_{\mu\nu},T_{\rho}]$ $\displaystyle=$
$\displaystyle\sigma(\mu\nu)_{\rho}^{\ \tau}T_{\tau},\ \
\,[T_{\mu\nu},\bar{T}_{\rho}]=\ -\sigma(\mu\nu)_{\rho}^{\
\tau}\bar{T}_{\tau},$ (8) $\displaystyle\,[M_{\mu\nu},T_{\rho}]$
$\displaystyle=$ $\displaystyle-{\,{\rm i}\,}\sigma[\mu\nu]_{\rho}^{\
\tau}T_{\tau},\ [M_{\mu\nu},\bar{T}_{\rho}]=-{\,{\rm
i}\,}\sigma[\mu\nu]_{\rho}^{\ \tau}\bar{T}_{\tau},$ (9)
$\displaystyle\,[M_{\mu\nu},T_{\rho\sigma}]$ $\displaystyle=$
$\displaystyle-{\,{\rm
i}\,}(\sigma[\mu\nu]_{\rho}^{\,\,\tau}\delta_{\sigma}^{\,\,\omega}+\sigma[\mu\nu]_{\sigma}^{\,\,\omega}\delta_{\rho}^{\tau})T_{\tau\omega},$
(10) $\displaystyle\,[M_{\mu\nu},M_{\rho\sigma}]$ $\displaystyle=$
$\displaystyle-{\,{\rm
i}\,}(\sigma[\mu\nu]_{\rho}^{\,\,\tau}\delta_{\sigma}^{\,\,\omega}+\sigma[\mu\nu]_{\sigma}^{\,\,\omega}\delta_{\rho}^{\tau})M_{\tau\omega},$
(11) $\displaystyle\,[T_{\mu\nu},T_{\rho\sigma}]$ $\displaystyle=$
$\displaystyle-{\,{\rm
i}\,}(\sigma(\mu\nu)_{\rho}^{\,\,\tau}\delta_{\sigma}^{\,\,\omega}-\sigma(\mu\nu)_{\sigma}^{\,\,\omega}\delta_{\rho}^{\tau})M_{\tau\omega},\
\ \rm{with}$ (12) $\displaystyle\sigma[\mu\nu]_{\rho}^{\tau}$
$\displaystyle:=$ $\displaystyle
g_{\mu\rho}\delta_{\nu}^{\,\,\sigma}-g_{\nu\rho}\delta_{\mu}^{\,\,\sigma},$
(13) $\displaystyle\sigma(\mu\nu)_{\rho}^{\tau}$ $\displaystyle:=$
$\displaystyle
g_{\mu\rho}\delta_{\nu}^{\,\,\sigma}+g_{\nu\rho}\delta_{\mu}^{\,\,\sigma}-(1/2)g_{\mu\nu}\delta_{\rho}^{\,\,\sigma},$
(14) $\displaystyle{\hat{T}_{[o]}}$ $\displaystyle:=$
$\displaystyle{T_{[o]}}+5{\cal T},$ (15)

where ${\cal T}$ is some central charge of this extension. The renormalization
carried out can absorb some constants, but sometimes is unable to modify a
sign. This is the origin of the parameter for $\epsilon\in\\{+1,-1\\}$.
Although we initially aimed it to be a sign, we could eventually let
$\epsilon\in\mathbb{R}$ as a free parameter that might decouple the obtained
extension. We need to track distinctly positive and negative powers of
$\epsilon$. We can not interpret ${\hat{T}_{[o]}}$ as a dilatation, but
$5{\cal T}={\hat{T}_{[o]}}-{T_{[o]}}$ – which is a central charge of this
subalgebra– could assume perhaps this role. The dimension 0 subalgebra
(5)–(15) with generators $T_{\mu},\hat{T}_{\mu},M_{\mu\nu},T_{\mu\nu}$,
$(24{T_{[o]}}+{\hat{T}_{[o]}})$ has dimension $24=5^{2}-1$ without its central
charge ${\hat{T}_{[o]}}-{T_{[o]}}$, and $25=5^{2}$ with it. This suggest a
possible relation respectively to the ${\mathfrak{su}}(t_{1},s_{1})$ and
${\mathfrak{u}}(t_{1},s_{1})$ with $s_{1},t_{1}>0$ and $s_{1}+t_{1}=5$. The
algebra of dimensionless generators (5)–(15) generated by
$T_{\mu},\hat{T}_{\mu},M_{\mu\nu},T_{\mu\nu}$, ${T_{[o]}},{\hat{T}_{[o]}}$
will be called the Lorentz Extension (LE). The reason for maintaining both
scalar generators ${T_{[o]}},{\hat{T}_{[o]}}$ is that the central charge
${\hat{T}_{[o]}}-{T_{[o]}}$ of this Lorentz extension is not anymore a central
charge when considering a Poincaré–like extension. And the selection of these
scalars as basis will assure that there exist quadratic casimir for both the
Lorentz and Poincaré extensions that does not mix both scalar generators
${T_{[o]}},{\hat{T}_{[o]}}$.

The subalgebra $l=<\\{M_{\mu\nu},T_{\rho\sigma}\\}>$ in (10)–(14) has
dimension $15=4^{2}-1=6\times 5/2$, which hints a possible relation to
${\mathfrak{su}}(t_{2},s_{2})$ with $s_{2},t_{2}>0$ and $s_{2}+t_{2}=4$, or to
${\mathfrak{so}}(t_{3},s_{3})$ with $s_{3},t_{3}>0$ and $s_{3}+t_{3}=6$ (the
maximal Lie algebra of symmetries of a De Sitter $t_{3}=1$ or anti–de Sitter
space $t_{3}=2$). Observe that there is a Klein geometry associated to this
subalgebra, since the complement to the standard Lorentz algebra in this
subalgebra $l$ is invariant under the Lorentz algebra, building a reductive
homogeneous space.

The subalgebra $<\\{M_{\mu\nu},{\cal P}_{\rho}\\}>$, with ${\cal P}_{\rho}$
being $(T_{\rho}+\hat{T}_{\rho})/\sqrt{2}$ or ${\,{\rm
i}\,}(T_{\rho}-\hat{T}_{\rho})/\sqrt{2}$, (notice $[{\cal P}_{\mu},{\cal
P}_{\nu}]={\,{\rm i}\,}\epsilon M_{\mu\nu}$) has dimension 10 and is not
isomorphic to the Poincaré algebra. It might be isomorphic to
${\mathfrak{so}}(t_{o},s_{o},)$ with $s_{o},t_{o}>0$ and $s_{o}+t_{o}=5$
(corresponding to a De Sitter $t_{o}=1$ for $\epsilon=-1$ or Anti–de Sitter
space $t_{o}=2$ for $\epsilon=1$). There is actually two copies of the
Poincaré algebra nested in the extension $<\\{M_{\mu\nu},T_{\rho}\\}>$, and
$<\\{M_{\mu\nu},\hat{T}_{\rho}\\}>$. We do not confer to the generators
$T_{\rho},\hat{T}_{\rho}$ the ultimate role of space-time translations, but
they might contribute as precursors of low energy translations. In fact, we
can consider a step towards contraction $T_{\rho}\mapsto\Lambda{\cal
T}_{\rho},\hat{T}_{\rho}\mapsto\Lambda\hat{\bf\cal T}_{\rho}$, whose scale
$\Lambda$ might or might not be connected to the one that produces the
contraction towards translations. We elaborate on this further under
alternative considerations.

The full dimensionless algebra, LE, has at least two casimirs, the central
charge ${\hat{T}_{[o]}}-{T_{[o]}}$, and a quadratic casimir $I$:

$\displaystyle I\\!\\!$ $\displaystyle:=$
$\displaystyle\\!\\!({2}/{\epsilon})\\{\hat{T}^{\omega},T_{\omega}\\}+M^{\rho\sigma}M_{\rho\sigma}+T^{\rho\sigma}T_{\rho\sigma}+(24/5){T_{[o]}}^{2}+(1/5)({\hat{T}_{[o]}})^{2}$
$\displaystyle=[{T}^{\omega}\ \hat{T}^{\omega}\ M^{\rho\tau}\ T^{\rho\tau}\
{T_{[o]}}\ {\hat{T}_{[o]}}]\cdot M\cdot[\hat{T}_{\omega}\ {T}_{\omega}\
M_{\rho\tau}\ T_{\rho\tau}\ {T_{[o]}}\ {\hat{T}_{[o]}}]^{tr}\\!\\!,\ \ $
$\displaystyle{\rm with}\ \ M:={\rm
diag}\left(\frac{2}{\epsilon},\frac{2}{\epsilon},1,1,\frac{24}{5},\frac{1}{5}\right)\
{\rm and}\
\\{\hat{T}^{\omega},T_{\omega}\\}:=\hat{T}^{\omega}T_{\omega}+T_{\omega}\hat{T}^{\omega}.$

In building this invariant, it intervenes both the lowering and rising of
Lorentz indices with $g_{\mu\nu}$ and the matrix $M$ depending on the type of
generator. Both contributions seem to build a Killing Metric. Whether such a
metric can be positive definite in some sense that allows for a well defined
vacuum state is not clear from the outset. We expect, that if the resulting
models give the right Lagrangian contributions terms for standard gauge and
scalar fields, there is some hope that a proper definition of a vacuum for the
theory might exist. This invariant actually suggested the adoption of
${\hat{T}_{[o]}}$ in (15) as the generator of a future would be internal
symmetry. The casimir (2) includes a portion $M^{\rho\sigma}M_{\rho\sigma}$
which is a casimir of the Lorentz subalgebra. There is actually two quadratic
casimirs for the semisimple Lorentz subalgebra. We do not know if there are
extra casimirs for the found extension (5)–(15) of the Lorentz algebra,
separating perhaps left–spin and right–spin representations. This extension
embraces not only the internal $U(1)$ symmetry generator ${T_{[o]}}$, but also
the external Lorentz symmetry generators $M_{\rho\sigma}$, a novel traceless
symmetric tensor (spin 2) generator $T_{\rho\sigma}$, and a would–be internal
symmetry generator ${\hat{T}_{[o]}}={T_{[o]}}+5{\cal T}$. But, each further
exploration will add to the understanding of these extensions.

We include now some comments on casimirs for some of the LE subalgebras. The
15 dimensional subalgebra $<\\{M_{\mu\nu},T_{\rho\sigma}\\}>$ has a quadratic
casimir

$I_{1}=M^{\mu\nu}M_{\mu\nu}+T^{\rho\sigma}T_{\rho\sigma}.$ (17)

Another quartic casimir is expected. There is subalgebras
$<\\{M_{\mu\nu},T_{\rho\sigma},T_{\omega}\\}>$ and
$<\\{M_{\mu\nu},T_{\rho\sigma},\hat{T}_{\omega}\\}>$ of dimension 19,
corresponding to inhomogeneous extensions of the 15 dimensional one. Now, the
subalgebra $<\\{M_{\mu\nu},{\cal P}_{\rho}\\}>$, with ${\cal P}_{\rho}$ being
$(T_{\rho}+\hat{T}_{\rho})/\sqrt{2}$ or ${\,{\rm
i}\,}(T_{\rho}-\hat{T}_{\rho})/\sqrt{2}$, generates a de Sitter
($\epsilon=-1$)/ Anti de Sitter ($\epsilon=1$) algebra. It has a quadratic
casimir

$I_{2}=\frac{1}{\epsilon}{\cal P}^{\rho}{\cal
P}_{\rho}+M^{\rho\sigma}M_{\rho\sigma}.$ (18)

A quartic casimir is obtained extrapolating on results of [8] to include
$\epsilon$:

$I_{3}=\frac{1}{\epsilon}\left(M^{\rho\sigma}M_{\rho\sigma}{\cal P}^{\mu}{\cal
P}_{\mu}-M^{\nu\sigma}{\cal P}_{\nu}M_{\mu\sigma}{\cal
P}_{\mu}\right)+\frac{1}{4}M^{\rho\sigma}M_{\rho\sigma}M^{\mu\nu}M_{\mu\nu}.$
(19)

We explore possible (pseudo scalar) invariants involving the levi-civita
tensor in Minkowski space. We find actually the following prospective casimir
that turns out to be just an identity. We call it a hollow casimir
(non–homogeneous identity):

$I_{4}=\epsilon^{\mu\nu\rho\sigma}\left(\frac{1}{\epsilon^{2}}{\cal
P}_{\mu}{\cal P}_{\nu}{\cal P}_{\rho}{\cal
P}_{\sigma}+M_{\mu\nu}M_{\rho\sigma}\right)=0.$ (20)

As we will see, these invariants, although trivial as these one, could be
instrumental to model certain portions of the lagrangian density of the fields
that embody the symmetry. In fact, for the full LE we can include the
following hollow casimir (non–homogneous identity):

$I_{5}=\epsilon^{\mu\nu\rho\sigma}\left({\frac{4}{\epsilon^{2}}}T_{\mu}\hat{T}_{\nu}T_{\rho}\hat{T}_{\sigma}+M_{\mu\nu}M_{\rho\sigma}\right)=0.$
(21)

##  3 Field representation, locally invariant model

We explore the field representation of the LE before we inquire about its
representation in terms of differential operators. We develop a formalism to
obtain what resembles a coadjoint representations on the fields of the
considered algebra, in this case for the dimensionless subalgebra (5)–(15). We
move to an interaction basis with generators ${T_{[o]}}_{int}:=h{T_{[o]}}$,
${\hat{T}_{[o]int}}:={\hat{h}}{\hat{T}_{[o]}},\ T_{\tau\
int}:=\ell\,T_{\tau},\ \hat{T}_{\tau\ int}:=\ell\,\hat{T}_{\tau},\
M_{\rho\sigma\ int}:={r^{\prime}}M_{\rho\sigma},T_{\rho\sigma\
int}:=rT_{\rho\sigma}$, but maintain the use of the original renormalized
algebra (5)–(15) and keep track of the coupling constants signaling an
interaction basis. We introduce several coupling constants in the expectation
that, after some symmetry breaking or a similar mechanism, each generator type
will evolve independently. To each dimensionless symmetry generator we
associate a field (connection) of naive dimension 1. We will develop the
covariant derivatives in a progressive fashion, starting with the standard
procedure. We will reserve from now on the symbols
$\alpha,\beta,\gamma,\delta$ for internal symmetry indices, the middle greek
letters $\varepsilon,\kappa,\eta$ for (Einstein) space–time indices, and
latter greek letters $\mu,\nu,\rho,\sigma,\tau,\omega,\delta$ for Lorentz
indices associated to the Lorentz extension. We propose first a covariant
derivative ${\cal D}_{\varepsilon}$:

$\displaystyle{\cal D}_{\varepsilon}:$ $\displaystyle=$
$\displaystyle\partial_{\varepsilon}+{\,{\rm
i}\,}hB_{\varepsilon}{T_{[o]}}+{\,{\rm
i}\,}\hat{h}\hat{B}_{\varepsilon}{\hat{T}_{[o]}}+{\,{\rm
i}\,}\ell(\hat{\Phi}_{\varepsilon}^{\,\,\tau}T_{\tau}+\Phi_{\varepsilon}^{\,\,\tau}\hat{T}_{\tau})$
(22) $\displaystyle+{\,{\rm
i}\,}r^{\prime}B^{\,\,[\rho\sigma]}_{\varepsilon}M_{\rho\sigma}+{\,{\rm
i}\,}rB^{\,\,(\rho\sigma)}_{\varepsilon}T_{\rho\sigma},$

where the fields
$B_{\varepsilon},\hat{B}_{\varepsilon},\hat{\Phi}_{\varepsilon}^{\,\,\tau},\Phi_{\varepsilon}^{\,\,\tau},B^{\,\,[\rho\sigma]}_{\varepsilon},B^{\,\,(\rho\sigma)}_{\varepsilon}$
are connections (gauge fields) corresponding respectively to the generators
${T_{[o]}},{\hat{T}_{[o]}},T_{\tau},\hat{T}_{\tau},M_{\rho\sigma},T_{\rho\sigma}$,
with respective coupling constants $h,\hat{h},\ell,\ell,r^{\prime},r$. Between
the connection and the symmetry generator we have a tensor product $\otimes$
which we also omit to alleviate notation. In place of $\partial_{\varepsilon}$
we would write $\partial_{\varepsilon}\otimes Id$, with $Id$ the universally
commuting identity operator. For each generator ${\cal O}$ and each covariant
derivative ${\cal D}_{\varepsilon}$ we define a generic quantum gauge field
${\cal C}_{\kappa}^{\ {\cal O}}$ and the generic field strength $\tilde{F}_{\
\varepsilon\kappa}^{{\cal O}}$:

$\tilde{F}_{\ \varepsilon\kappa}^{{\cal O}}:=\partial_{\varepsilon}{\cal
C}_{\kappa}^{\ {\cal O}}-\partial_{\kappa}{\cal C}_{\varepsilon}^{\ {\cal
O}}+{\cal F}_{\ \varepsilon\kappa}^{{\cal O}},$ (23)

where ${\cal F}_{\ \varepsilon\kappa}^{{\cal O}}$ is quadratic in the fields.
The formal commutator $[{\cal D}_{\varepsilon},{\cal D}_{\kappa}]$ is expanded
in terms of the field strengths with the notation

$\displaystyle\,[{\cal D}_{\varepsilon},{\cal D}_{\kappa}]\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!{\,{\rm
i}\,}(\tilde{F}_{\varepsilon\kappa})h\,{T_{[o]}}+{\,{\rm
i}\,}({\tilde{\hat{F}}}_{\varepsilon\kappa})\hat{h}\,{\hat{T}_{[o]}}+{\,{\rm
i}\,}(\tilde{\hat{{F}}}_{\ \varepsilon\kappa}^{\rho})\ell\,T_{\rho}+{\,{\rm
i}\,}(\tilde{F}_{\ \varepsilon\kappa}^{\rho})\ell\,\hat{T}_{\rho}$ (24)
$\displaystyle\\!\\!+{\,{\rm i}\,}(\tilde{G}_{\ \ \
\varepsilon\kappa}^{[\tau\omega]})r^{\prime}\,M_{\tau\omega}+{\,{\rm
i}\,}(\tilde{H}_{\ \ \ \varepsilon\kappa}^{(\tau\omega)})r\,T_{\tau\omega}.$

Actual computation using the interaction basis leads to quadratic parts ${\cal
F}_{\ \varepsilon\kappa}^{{\cal O}}$ associated respectively to the generators
${T_{[o]}},{\hat{T}_{[o]}},T_{\tau},\hat{T}_{\tau},M_{\rho\sigma},T_{\rho\sigma}$:

$\displaystyle{\cal F}_{\varepsilon\kappa}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!-\\!{\,{\rm
i}\,}\hat{\Phi}_{\varepsilon}^{\,\,\rho}\\!\left(\\!-\frac{12\epsilon\ell^{2}}{5h}g_{\rho\sigma}\\!\right)\\!\Phi_{\kappa}^{\,\,\sigma}\\!-{\,{\rm
i}\,}\Phi_{\varepsilon}^{\,\,\rho}\\!\left(\\!+\frac{12\epsilon\ell^{2}}{5h}g_{\rho\sigma}\\!\right)\\!\hat{\Phi}_{\kappa}^{\,\,\sigma}\\!,$
(25) $\displaystyle\hat{{\cal F}}_{\varepsilon\kappa}\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!-\\!{\,{\rm
i}\,}\hat{\Phi}_{\varepsilon}^{\,\,\rho}\\!\left(\\!-\frac{\epsilon\ell^{2}}{10\hat{h}}g_{\rho\sigma}\\!\right)\\!\Phi_{\kappa}^{\,\,\sigma}-\\!{\,{\rm
i}\,}\Phi_{\varepsilon}^{\,\,\rho}\\!\left(\\!+\frac{\epsilon\ell^{2}}{10\hat{h}}g_{\rho\sigma}\\!\right)\\!\hat{\Phi}_{\kappa}^{\,\,\sigma},$
(26) $\displaystyle{\cal F}_{\ \varepsilon\kappa}^{\rho}\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!-{\,{\rm
i}\,}B_{\varepsilon}(+h)\Phi_{\kappa}^{\,\,\rho}-{\,{\rm
i}\,}\Phi_{\varepsilon}^{\,\,\rho}(-h)B_{\kappa}-{\,{\rm
i}\,}\hat{B}_{\varepsilon}(+\hat{h})\Phi_{\kappa}^{\,\,\rho}-{\,{\rm
i}\,}\Phi_{\varepsilon}^{\,\,\rho}(-\hat{h})\hat{B}_{\kappa}$ (27)
$\displaystyle\hskip 31.2982pt-{\,{\rm
i}\,}B_{\varepsilon}^{\,\,[\tau\omega]}(+{\,{\rm
i}\,}r^{\prime}\sigma[\tau\omega]_{\sigma}^{\,\,\rho})\Phi_{\kappa}^{\,\,\sigma}-{\,{\rm
i}\,}\Phi_{\varepsilon}^{\,\,\sigma}(-{\,{\rm
i}\,}r^{\prime}\sigma[\tau\omega]_{\sigma}^{\,\,\rho})B_{\kappa}^{\,\,[\tau\omega]}$
$\displaystyle\hskip 31.2982pt-{\,{\rm
i}\,}B_{\varepsilon}^{\,\,(\tau\omega)}(+r\sigma(\tau\omega)_{\sigma}^{\,\,\rho})\Phi_{\kappa}^{\,\,\sigma}-{\,{\rm
i}\,}\Phi_{\varepsilon}^{\,\,\sigma}(-r\sigma(\tau\omega)_{\sigma}^{\,\,\rho})B_{\kappa}^{\,\,(\tau\omega)},$
$\displaystyle\hat{{\cal F}}_{\ \varepsilon\kappa}^{\rho}\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!-{\,{\rm
i}\,}B_{\varepsilon}(-h)\hat{\Phi}_{\kappa}^{\,\,\rho}-{\,{\rm
i}\,}\hat{\Phi}_{\varepsilon}^{\,\,\rho}(+h)B_{\kappa}-{\,{\rm
i}\,}\hat{B}_{\varepsilon}(-\hat{h})\hat{\Phi}_{\kappa}^{\,\,\rho}-{\,{\rm
i}\,}\hat{\Phi}_{\varepsilon}^{\,\,\rho}(+\hat{h})\hat{B}_{\kappa}$ (28)
$\displaystyle\hskip 31.2982pt-{\,{\rm
i}\,}B_{\varepsilon}^{\,\,[\tau\omega]}(+{\,{\rm
i}\,}r^{\prime}\sigma[\tau\omega]_{\sigma}^{\,\,\rho})\hat{\Phi}_{\kappa}^{\,\,\sigma}-{\,{\rm
i}\,}\hat{\Phi}_{\varepsilon}^{\,\,\sigma}(-{\,{\rm
i}\,}r^{\prime}\sigma[\tau\omega]_{\sigma}^{\,\,\rho})B_{\kappa}^{\,\,[\tau\omega]}$
$\displaystyle\hskip 31.2982pt-{\,{\rm
i}\,}B_{\varepsilon}^{\,\,(\tau\omega)}(-r\sigma(\tau\omega)_{\sigma}^{\,\,\rho})\hat{\Phi}_{\kappa}^{\,\,\sigma}-{\,{\rm
i}\,}\hat{\Phi}_{\varepsilon}^{\,\,\sigma}(+r\sigma(\tau\omega)_{\sigma}^{\,\,\rho})B_{\kappa}^{\,\,(\tau\omega)},$
$\displaystyle{\cal G}_{\ \ \ \varepsilon\kappa}^{[\tau\omega]}\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!-\\!{\,{\rm
i}\,}\hat{\Phi}_{\varepsilon}^{\,\,\mu}\\!\\!\left(\\!+\frac{{\,{\rm
i}\,}\epsilon\ell^{2}}{2r^{\prime}}(\delta_{\mu}^{\,\,\tau}\delta_{\nu}^{\,\,\omega}\\!-\\!\delta_{\mu}^{\,\,\omega}\delta_{\nu}^{\,\,\tau})\\!\right)\\!\\!\Phi_{\kappa}^{\,\,\nu}\\!-\\!{\,{\rm
i}\,}\Phi_{\varepsilon}^{\,\,\mu}\\!\\!\left(\\!+\frac{{\,{\rm
i}\,}\epsilon\ell^{2}}{2r^{\prime}}(\delta_{\mu}^{\,\,\tau}\delta_{\nu}^{\,\,\omega}\\!-\\!\delta_{\mu}^{\,\,\omega}\delta_{\nu}^{\,\,\tau})\\!\right)\\!\\!\hat{\Phi}_{\kappa}^{\,\,\nu}\\!\\!$
(29) $\displaystyle\hskip 31.2982pt-{\,{\rm
i}\,}B_{\varepsilon}^{\,\,[\rho\sigma]}(+{\,{\rm
i}\,}r^{\prime}\left(\sigma[\rho\sigma]_{\mu}^{\,\,\tau}\delta_{\nu}^{\,\,\omega}+\sigma[\rho\sigma]_{\nu}^{\,\,\omega}\delta_{\mu}^{\,\,\tau}\right))B_{\kappa}^{\,\,[\mu\nu]}$
$\displaystyle\hskip 31.2982pt-{\,{\rm
i}\,}B_{\varepsilon}^{\,\,(\rho\sigma)}\left(+\frac{{\,{\rm
i}\,}r^{2}}{r^{\prime}}\left(\sigma(\rho\sigma)_{\mu}^{\,\,\tau}\delta_{\nu}^{\,\,\omega}-\sigma(\rho\sigma)_{\nu}^{\,\,\omega}\delta_{\mu}^{\,\,\tau}\right)\right)B_{\kappa}^{\,\,(\mu\nu)},$
$\displaystyle\ {\cal H}_{\ \ \ \varepsilon\kappa}^{(\tau\omega)}\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!-{\,{\rm
i}\,}B_{\varepsilon}^{\,\,[\rho\sigma]}(+{\,{\rm
i}\,}r^{\prime}\left(\sigma[\rho\sigma]_{\mu}^{\,\,\tau}\delta_{\nu}^{\,\,\omega}+\sigma[\rho\sigma]_{\nu}^{\,\,\omega}\delta_{\mu}^{\,\,\tau}\right))B_{\kappa}^{\,\,(\mu\nu)}$
(30) $\displaystyle\hskip 31.2982pt-{\,{\rm
i}\,}B_{\varepsilon}^{\,\,(\mu\nu)}(-{\,{\rm
i}\,}r^{\prime}\left(\sigma(\mu\nu)_{\rho}^{\,\,\tau}\delta_{\sigma}^{\,\,\omega}-\sigma(\mu\nu)_{\sigma}^{\,\,\omega}\delta_{\rho}^{\,\,\tau}\right))B_{\kappa}^{\,\,[\rho\sigma]}$
$\displaystyle\hskip 31.2982pt-\\!{\,{\rm
i}\,}\hat{\Phi}_{\varepsilon}^{\,\,\mu}\\!\left(\\!-\frac{\epsilon\ell^{2}}{2r}\left(\delta_{\mu}^{\,\,\tau}\delta_{\nu}^{\,\,\omega}+\delta_{\mu}^{\,\,\omega}\delta_{\nu}^{\,\,\tau}-\mbox{$\frac{1}{2}$}g^{\tau\omega}g_{\mu\nu}\right)\\!\right)\\!\Phi_{\kappa}^{\,\,\nu}$
$\displaystyle\hskip 31.2982pt-{\,{\rm
i}\,}\Phi_{\varepsilon}^{\,\,\mu}\\!\left(\\!+\frac{\epsilon\ell^{2}}{2r}\left(\delta_{\mu}^{\,\,\tau}\delta_{\nu}^{\,\,\omega}+\delta_{\mu}^{\,\,\omega}\delta_{\nu}^{\,\,\tau}-\mbox{$\frac{1}{2}$}g^{\tau\omega}g_{\mu\nu}\right)\\!\right)\\!\hat{\Phi}_{\kappa}^{\,\,\nu}.$

We adopt thus the generic Ansatz

$\displaystyle\\!\lambda^{{\cal O}}{\cal F}_{\ \varepsilon\kappa}^{{\cal
O}}\\!=\\!-{{\,{\rm i}\,}}\sum_{{\cal O}^{\prime\prime}}\\!\\!\left({\cal
C}_{\varepsilon}^{\ {\cal O}^{\prime\prime}}\lambda^{{\cal
O}^{\prime\prime}}[{\cal O}^{\prime\prime},\lambda^{{\cal O}}{\cal
C}_{\kappa}^{\ {\cal O}}]_{1}\right)\\!=\\!{{\,{\rm i}\,}}(\lambda^{{\cal
O}^{\prime\prime}}{\cal C}_{\varepsilon}^{\ {\cal O}^{\prime\prime}}){\,{\rm
i}\,}C_{\ {\cal O}^{\prime\prime}{\cal O}^{\prime}}^{{\cal O}}(\lambda^{{\cal
O}^{\prime}}{\cal C}_{\varepsilon}^{\ {\cal O}^{\prime}}),$ (31)
$\displaystyle\,[{\cal O}^{\prime\prime},\lambda^{{\cal O}}{\cal
C}_{\varepsilon}^{\ {\cal O}}]_{1}\\!=\\!-{\,{\rm i}\,}C_{\ {\cal
O}^{\prime\prime}{\cal O}^{\prime}}^{{\cal O}}(\lambda^{{\cal
O}^{\prime}}{\cal C}_{\varepsilon}^{\ {\cal O}^{\prime}}),\ {\rm for}\ \
[{\cal O}^{\prime\prime},{\cal O}^{\prime}]={\,{\rm i}\,}\,C_{\ {\cal
O}^{\prime\prime}{\cal O}^{\prime}}^{{\cal O}}\,{\cal O}.$ (32)

From it, we can read the field transformations, for
$b=\varepsilon\in\\{0,1,2,3\\}$:

$\displaystyle\,[T_{\mu},hB_{b}]_{1}$ $\displaystyle=$
$\displaystyle\\!\left(\\!-\frac{12\epsilon\ell}{5}g_{\mu\sigma}\\!\right)\\!\Phi_{b}^{\,\,\sigma},\hskip
8.5359pt[\hat{T}_{\mu},hB_{b}]_{1}\ =\
\\!\left(\\!+\frac{12\epsilon\ell}{5}g_{\mu\sigma}\\!\right)\\!\hat{\Phi}_{b}^{\,\,\sigma},$
(33) $\displaystyle\,[T_{\mu},\hat{h}\hat{B}_{b}]_{1}$ $\displaystyle=$
$\displaystyle\\!\left(\\!-\frac{\epsilon\ell}{10}g_{\mu\sigma}\\!\right)\\!\Phi_{b}^{\,\,\sigma},\hskip
17.07182pt[\hat{T}_{\mu},\hat{h}\hat{B}_{b}]_{1}\ =\
\\!\left(\\!+\frac{\epsilon\ell}{10}g_{\mu\sigma}\\!\right)\\!\hat{\Phi}_{b}^{\,\,\sigma},$
(34) $\displaystyle\,[T_{\mu},r^{\prime}B_{b}^{\,\,[\tau\kappa]}]_{1}$
$\displaystyle=$ $\displaystyle\\!\left(\\!+\frac{{\,{\rm
i}\,}\epsilon\ell}{2}(\delta_{\mu}^{\,\,\tau}\delta_{\delta}^{\,\,\kappa}-\delta_{\mu}^{\,\,\kappa}\delta_{\delta}^{\,\,\tau})\\!\right)\\!\Phi_{b}^{\,\,\delta},$
(35)
$\displaystyle\\!\\!\\![\hat{T}_{\mu},r^{\prime}B_{b}^{\,\,[\tau\kappa]}]_{1}\
=\ \\!\\!\left(\\!+\frac{{\,{\rm
i}\,}\epsilon\ell}{2}(\delta_{\mu}^{\,\,\tau}\delta_{\delta}^{\,\,\kappa}-\delta_{\mu}^{\,\,\kappa}\delta_{\delta}^{\,\,\tau})\\!\right)\\!\hat{\Phi}_{b}^{\,\,\delta},$
$\displaystyle\,[T_{\mu},rB_{b}^{\,\,(\tau\kappa)}]_{1}$ $\displaystyle=$
$\displaystyle\\!\left(\\!-\frac{\epsilon\ell}{2}(\delta_{\mu}^{\,\,\tau}\delta_{\delta}^{\,\,\kappa}+\delta_{\mu}^{\,\,\kappa}\delta_{\delta}^{\,\,\tau}\\!-\mbox{$\frac{1}{2}$}g^{\tau\kappa}g_{\mu\delta})\\!\right)\\!\Phi_{b}^{\,\,\delta},$
(36) $\displaystyle\\!\\!\\![\hat{T}_{\mu},rB_{b}^{\,\,(\tau\kappa)}]_{1}=\
\\!\\!\left(\\!\frac{\epsilon\ell}{2}(\delta_{\mu}^{\,\,\tau}\delta_{\delta}^{\,\,\kappa}\\!+\\!\delta_{\mu}^{\,\,\kappa}\delta_{\delta}^{\,\,\tau}\\!-\\!\mbox{$\frac{1}{2}$}g^{\tau\kappa}g_{\mu\delta})\\!\right)\\!\hat{\Phi}_{b}^{\,\,\delta},\
\ \ $ $\displaystyle\,[T_{\mu},\ell\,\Phi_{b}^{\,\,\tau}]$ $\displaystyle=$
$\displaystyle 0,\hskip
82.51282pt[\hat{T}_{\mu},\ell\,\hat{\Phi}_{b}^{\,\,\tau}]_{1}\ =\ 0,$ (37)
$\displaystyle\,[T_{\mu},\ell\,\hat{\Phi}_{b}^{\,\,\tau}]_{1}$
$\displaystyle=$
$\displaystyle\\!h\delta_{\mu}^{\,\,\tau}B_{b}+\hat{h}\delta_{\mu}^{\,\,\tau}\hat{B}_{b}-\\!{\,{\rm
i}\,}r^{\prime}\sigma[\rho\sigma]_{\mu}^{\,\,\tau}B_{b}^{\,\,[\rho\sigma]}+r\sigma(\rho\sigma)_{\mu}^{\,\,\tau}\\!B_{b}^{\,\,(\rho\sigma)}\\!\\!\\!,$
$\displaystyle\,[\hat{T}_{\mu},\ell\,\Phi_{b}^{\,\,\tau}]_{1}$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!-h\delta_{\mu}^{\,\,\tau}B_{b}-\hat{h}\delta_{\mu}^{\,\,\tau}\hat{B}_{b}-\\!{\,{\rm
i}\,}r^{\prime}\sigma[\rho\sigma]_{\mu}^{\,\,\tau}B_{b}^{\,\,[\rho\sigma]}-r\sigma(\rho\sigma)_{\mu}^{\,\,\tau}\\!B_{b}^{\,\,(\rho\sigma)}\\!\\!\\!,\
\ \ $ (38) $\displaystyle\,[{T_{[o]}},hB_{b}]_{1}$ $\displaystyle=$
$\displaystyle 0,\hskip 82.51282pt[{T_{[o]}},\hat{h}\hat{B}_{b}]_{1}\ =\ 0,$
(39) $\displaystyle\,[{T_{[o]}},r^{\prime}B_{b}^{\,\,[\tau\kappa]}]_{1}$
$\displaystyle=$ $\displaystyle 0,\hskip
68.28644pt[{T_{[o]}},rB_{b}^{\,\,(\tau\kappa)}]_{1}\ =\ 0,$ (40)
$\displaystyle\,[{T_{[o]}},\ell\,\Phi_{b}^{\,\,\tau}]_{1}$ $\displaystyle=$
$\displaystyle+\ell\,\Phi_{b}^{\,\,\tau},\hskip
56.9055pt[{T_{[o]}},\ell\,\hat{\Phi}_{b}^{\,\,\tau}]_{1}\
=-\ell\,\hat{\Phi}_{b}^{\,\,\tau},$ (41)
$\displaystyle\,[{\hat{T}_{[o]}},hB_{b}]_{1}$ $\displaystyle=$ $\displaystyle
0,\hskip 82.51282pt[{\hat{T}_{[o]}},\hat{h}\,\hat{B}_{b}]_{1}\ =\ 0,$ (42)
$\displaystyle\\!\\!\\![{\hat{T}_{[o]}},r^{\prime}B_{b}^{\,\,[\tau\kappa]}]_{1}$
$\displaystyle=$ $\displaystyle 0,\hskip
68.28644pt[{\hat{T}_{[o]}},rB_{b}^{\,\,(\tau\kappa)}]_{1}\ =\ 0,$ (43)
$\displaystyle\,[{\hat{T}_{[o]}},\ell\,\Phi_{b}^{\,\,\tau}]_{1}$
$\displaystyle=$ $\displaystyle+\ell\,\Phi_{b}^{\,\,\tau},\hskip
56.9055pt[{\hat{T}_{[o]}},\ell\,\hat{\Phi}_{b}^{\,\,\tau}]_{1}\
=-\ell\,\hat{\Phi}_{b}^{\,\,\tau},$ (44)
$\displaystyle\,[M_{\rho\sigma},hB_{b}]_{1}$ $\displaystyle=$ $\displaystyle
0,\hskip 82.51282pt[M_{\rho\sigma},\hat{h}\hat{B}_{b}]_{1}=\ 0,$ (45)
$\displaystyle\\!\\!\\![M_{\rho\sigma},r^{\prime}B_{b}^{\,\,[\tau\kappa]}]_{1}$
$\displaystyle=$ $\displaystyle+{\,{\rm
i}\,}r^{\prime}(\sigma[\rho\sigma]_{\gamma}^{\,\,\tau}\delta_{\delta}^{\,\,\kappa}+\sigma[\rho\sigma]_{\delta}^{\,\,\kappa}\delta_{\gamma}^{\tau})B_{b}^{\,\,[\gamma\delta]},$
(46) $\displaystyle\\!\\!\\![M_{\rho\sigma},rB_{b}^{\,\,(\tau\kappa)}]_{1}$
$\displaystyle=$ $\displaystyle+{\,{\rm
i}\,}r(\sigma[\rho\sigma]_{\gamma}^{\,\,\tau}\delta_{\delta}^{\,\,\kappa}+\sigma[\rho\sigma]_{\delta}^{\,\,\kappa}\delta_{\gamma}^{\tau})B_{b}^{\,\,(\gamma\delta)},$
(47) $\displaystyle\,[M_{\rho\sigma},\ell\,\Phi_{b}^{\,\,\tau}]_{1}$
$\displaystyle=$ $\displaystyle+{\,{\rm
i}\,}\ell\,\sigma[\rho\sigma]_{\gamma}^{\,\,\tau}\Phi_{b}^{\,\,\gamma},\hskip
5.69046pt[M_{\rho\sigma},\ell\,\hat{\Phi}_{b}^{\,\,\tau}]_{1}\ =\ +{\,{\rm
i}\,}\ell\,\sigma[\rho\sigma]_{\gamma}^{\,\,\tau}\hat{\Phi}_{b}^{\,\,\gamma},\
$ (48) $\displaystyle\\!\\![T_{\rho\sigma},hB_{b}]_{1}$ $\displaystyle=$
$\displaystyle 0,\hskip 82.51282pt[T_{\rho\sigma},\hat{h}\hat{B}_{b}]_{1}=\
0,$ (49)
$\displaystyle\,[T_{\rho\sigma},r^{\prime}B_{b}^{\,\,[\tau\kappa]}]_{1}$
$\displaystyle=$ $\displaystyle+{\,{\rm
i}\,}r(\sigma(\rho\sigma)_{\gamma}^{\,\,\tau}\delta_{\delta}^{\,\,\kappa}-\sigma(\rho\sigma)_{\delta}^{\,\,\kappa}\delta_{\gamma}^{\,\,\tau})B_{b}^{\,\,(\gamma\delta)},$
(50) $\displaystyle\\!\\![T_{\rho\sigma},rB_{b}^{\,\,(\tau\kappa)}]_{1}$
$\displaystyle=$ $\displaystyle+{\,{\rm
i}\,}r^{\prime}(\sigma(\rho\sigma)_{\gamma}^{\,\,\tau}\delta_{\delta}^{\,\,\kappa}-\sigma(\rho\sigma)_{\delta}^{\,\,\kappa}\delta_{\gamma}^{\,\,\tau})B_{b}^{\,\,[\gamma\delta]},$
(51) $\displaystyle\,[T_{\rho\sigma},\ell\,\Phi_{b}^{\,\,\tau}]_{1}$
$\displaystyle=$
$\displaystyle+\ell\,\sigma(\rho\sigma)_{\gamma}^{\,\,\tau}\Phi_{b}^{\,\,\gamma},\hskip
17.07182pt[T_{\rho\sigma},\ell\,\hat{\Phi}_{b}^{\,\,\tau}]_{1}\ =\
-\ell\,\sigma(\rho\sigma)_{\gamma}^{\,\,\tau}\hat{\Phi}_{b}^{\,\,\gamma}.$
(52)

It is easy to verify that the connections
$hB_{\kappa},\hat{h}\hat{B}_{\kappa},\ell\,\hat{\Phi}_{\kappa}^{\,\,\tau},\ell\,\Phi_{\kappa}^{\,\,\tau},r^{\prime}B^{\,\,[\tau\omega]}_{\kappa},rB^{\,\,(\tau\omega)}_{\kappa}$
with $[{\cal O},\cdot]_{1}$ build a left lie–algebra module, that is, it is a
representation of the LE algebra (5)–(15). Notice that the subindex $b=\kappa$
is an Einstein space–time label which is not transformed, nor affected, by the
considered transformations. This subindex is attached to the local space–time
coordinate derivative which is assumed thus far to commute with all symmetry
transformations. This is obviously not necessarily the case. Further
elaborations will address the transformation of geometric or coordinate
space–time frame in a further section.

The first order gauge transformations $({\cal C}_{\mu}^{\ {\cal O}})^{\prime}$
with space–time point dependent parameters $\varsigma(x),\hat{\varsigma}(x)$,
$\varsigma^{[\tau\omega]}(x),\varsigma^{(\tau\omega)}(x),\hat{\varsigma}^{\nu}(x),\varsigma^{\nu}(x)$
associated respectively to transformations with generators
${T_{[o]}},\bar{T}_{[o]},M_{\tau\omega},T_{\tau\omega},T_{\nu},\hat{T}_{\nu}$
is given by:

$\displaystyle(B_{\kappa})^{\prime}$ $\displaystyle=$ $\displaystyle
B_{\kappa}+{\,{\rm
i}\,}\frac{\hat{\varsigma}^{\nu}(x)}{\ell}\left(\\!-\frac{12\epsilon\ell^{2}}{5h}g_{\nu\sigma}\\!\right)\\!\Phi_{\kappa}^{\,\,\sigma}$
(53) $\displaystyle\hskip 11.38092pt+{\,{\rm
i}\,}\frac{\varsigma^{\nu}(x)}{\ell}\left(\\!+\frac{12\epsilon\ell^{2}}{5h}g_{\nu\sigma}\\!\right)\\!\hat{\Phi}_{\kappa}^{\,\,\sigma}+\frac{1}{h}\partial_{\kappa}\varsigma(x),$
$\displaystyle(\hat{B}_{\kappa})^{\prime}$ $\displaystyle=$
$\displaystyle\hat{B}_{\kappa}+{\,{\rm
i}\,}\frac{\hat{\varsigma}^{\nu}(x)}{\ell}\left(\\!-\frac{\epsilon\ell^{2}}{10\hat{h}}g_{\nu\sigma}\\!\right)\\!\Phi_{\kappa}^{\,\,\sigma}$
(54) $\displaystyle\hskip 11.38092pt+{\,{\rm
i}\,}\frac{\varsigma^{\nu}(x)}{\ell}\left(\\!+\frac{\epsilon\ell^{2}}{10\hat{h}}g_{\nu\sigma}\\!\right)\\!\hat{\Phi}_{\kappa}^{\,\,\sigma}+\frac{1}{\hat{h}}\partial_{\kappa}\hat{\varsigma}(x),$
$\displaystyle(\Phi_{\kappa}^{\,\,\rho})^{\prime}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\Phi_{\kappa}^{\,\,\rho}+{\,{\rm
i}\,}\frac{\varsigma^{\sigma}(x)}{\ell}\\!\left(\\!-h\delta_{\sigma}^{\,\,\rho}B_{\kappa}\\!-\hat{h}\delta_{\sigma}^{\,\,\rho}\hat{B}_{\kappa}\\!-\\!{\,{\rm
i}\,}r^{\prime}\sigma[\tau\omega]_{\sigma}^{\,\,\rho}B_{\kappa}^{\,\,[\tau\omega]}-r\sigma(\tau\omega)_{\sigma}^{\,\,\rho}B_{\kappa}^{\,\,(\tau\omega)}\\!\right)$
(55) $\displaystyle+{\,{\rm
i}\,}\frac{\varsigma^{[\tau\omega]}(x)}{r^{\prime}}(+{\,{\rm
i}\,}r^{\prime}\sigma[\tau\omega]_{\sigma}^{\,\,\rho})\Phi_{\kappa}^{\,\,\sigma}+{\,{\rm
i}\,}\frac{\varsigma^{(\tau\omega)}(x)}{r}(+r\sigma(\tau\omega)_{\sigma}^{\,\,\rho})\Phi_{\kappa}^{\,\,\sigma}$
$\displaystyle+{\,{\rm
i}\,}\frac{\varsigma(x)}{h}(+h)\Phi_{\kappa}^{\,\,\rho}+{\,{\rm
i}\,}\frac{\hat{\varsigma}(x)}{\hat{h}}(+\hat{h})\Phi_{\kappa}^{\,\,\rho}\\!+\\!\frac{1}{\ell}\partial_{\kappa}\varsigma^{\rho}(x),$
$\displaystyle(\hat{\Phi}_{\kappa}^{\,\,\rho})^{\prime}\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\hat{\Phi}_{\kappa}^{\,\,\rho}+{\,{\rm
i}\,}\frac{\hat{\varsigma}^{\sigma}(x)}{\ell}\\!\left(\\!h\delta_{\sigma}^{\,\,\rho}B_{\kappa}\\!+\hat{h}\delta_{\sigma}^{\,\,\rho}\hat{B}_{\kappa}\\!-\\!{\,{\rm
i}\,}r^{\prime}\sigma[\tau\omega]_{\sigma}^{\,\,\rho}B_{\kappa}^{\,\,[\tau\omega]}+r\sigma(\tau\omega)_{\sigma}^{\,\,\rho}B_{\kappa}^{\,\,(\tau\omega)}\\!\right)$
(56) $\displaystyle+{\,{\rm
i}\,}\frac{\varsigma^{[\tau\omega]}(x)}{r^{\prime}}(+{\,{\rm
i}\,}r^{\prime}\sigma[\tau\omega]_{\sigma}^{\,\,\rho})\hat{\Phi}_{\kappa}^{\,\,\sigma}+{\,{\rm
i}\,}\frac{\varsigma^{(\tau\omega)}(x)}{r}(-r\sigma(\tau\omega)_{\sigma}^{\,\,\rho})\hat{\Phi}_{\kappa}^{\,\,\sigma}$
$\displaystyle+{\,{\rm
i}\,}\frac{\varsigma(x)}{h}(-h)\hat{\Phi}_{\kappa}^{\,\,\rho}+{\,{\rm
i}\,}\frac{\hat{\varsigma}(x)}{\hat{h}}(-\hat{h})\hat{\Phi}_{\kappa}^{\,\,\rho}\\!+\\!\frac{1}{\ell}\partial_{\kappa}\hat{\varsigma}^{\rho}(x),$
$\displaystyle\\!\\!\\!(B_{\kappa}^{\,\,[\tau\omega]})^{\prime}\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!B_{\kappa}^{\,\,[\tau\omega]}+{\,{\rm
i}\,}\frac{\varsigma^{[\rho\sigma]}(x)}{r^{\prime}}(+{\,{\rm
i}\,}r^{\prime}\left(\sigma[\rho\sigma]_{\mu}^{\,\,\tau}\delta_{\nu}^{\,\,\omega}+\sigma[\rho\sigma]_{\nu}^{\,\,\omega}\delta_{\mu}^{\,\,\tau}\right))B_{\kappa}^{\,\,[\mu\nu]}$
(57) $\displaystyle\hskip 11.38092pt+{\,{\rm
i}\,}\frac{{\varsigma}^{(\rho\sigma)}(x)}{r}\left(-\frac{{\,{\rm
i}\,}r^{2}}{r^{\prime}}\left(\sigma(\rho\sigma)_{\mu}^{\,\,\tau}\delta_{\nu}^{\,\,\omega}-\sigma(\rho\sigma)_{\nu}^{\,\,\omega}\delta_{\mu}^{\,\,\tau}\right)\right)B_{\kappa}^{\,\,(\mu\nu)}$
$\displaystyle\hskip 11.38092pt+\\!{\,{\rm
i}\,}\frac{\hat{\varsigma}^{\mu}(x)}{\ell}\\!\left(\\!+\frac{{\,{\rm
i}\,}\epsilon\ell^{2}}{2r^{\prime}}(\delta_{\mu}^{\,\,\tau}\delta_{\nu}^{\,\,\omega}-\delta_{\mu}^{\,\,\omega}\delta_{\nu}^{\,\,\tau})\\!\right)\\!\Phi_{\kappa}^{\,\,\nu}$
$\displaystyle\hskip 11.38092pt+{\,{\rm
i}\,}\frac{\varsigma^{\mu}(x)}{\ell}\\!\left(\\!+\frac{{\,{\rm
i}\,}\epsilon\ell^{2}}{2r^{\prime}}(\delta_{\mu}^{\,\,\tau}\delta_{\nu}^{\,\,\omega}-\delta_{\mu}^{\,\,\omega}\delta_{\nu}^{\,\,\tau})\\!\right)\\!\hat{\Phi}_{\kappa}^{\,\,\nu}+\frac{1}{r^{\prime}}\partial_{\kappa}\varsigma^{[\tau\omega]}(x),$
$\displaystyle\\!\\!\\!(B_{\kappa}^{\,\,(\tau\omega)})^{\prime}\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!B_{\kappa}^{\,\,(\tau\omega)}+{\,{\rm
i}\,}\frac{\varsigma^{[\rho\sigma]}(x)}{r^{\prime}}(+{\,{\rm
i}\,}r^{\prime}\left(\sigma[\rho\sigma]_{\mu}^{\,\,\tau}\delta_{\nu}^{\,\,\omega}+\sigma[\rho\sigma]_{\nu}^{\,\,\omega}\delta_{\mu}^{\,\,\tau}\right))B_{\kappa}^{\,\,(\mu\nu)}$
(58) $\displaystyle\hskip 11.38092pt+{\,{\rm
i}\,}\frac{{\varsigma}^{(\rho\sigma)}(x)}{r}(-{\,{\rm
i}\,}r^{\prime}\left(\sigma[\mu\nu]_{\rho}^{\,\,\tau}\delta_{\sigma}^{\,\,\omega}+\sigma[\mu\nu]_{\sigma}^{\,\,\omega}\delta_{\rho}^{\,\,\tau}\right))B_{\kappa}^{\,\,[\mu\nu]}$
$\displaystyle\hskip 11.38092pt+{\,{\rm
i}\,}\frac{\hat{\varsigma}^{\mu}(x)}{\ell}\\!\left(\\!-\frac{\epsilon\ell^{2}}{2r}\left(\delta_{\mu}^{\,\,\tau}\delta_{\nu}^{\,\,\omega}\\!+\\!\delta_{\mu}^{\,\,\omega}\delta_{\nu}^{\,\,\tau}\\!-\\!\mbox{$\frac{1}{2}$}g^{\tau\omega}g_{\mu\nu}\right)\\!\right)\\!\Phi_{\kappa}^{\,\,\nu}$
$\displaystyle\\!\\!\\!+{\,{\rm
i}\,}\frac{\varsigma^{\mu}(x)}{\ell}\\!\left(\\!+\\!\frac{\epsilon\ell^{2}}{2r}\left(\delta_{\mu}^{\,\,\tau}\delta_{\nu}^{\,\,\omega}\\!\\!+\\!\delta_{\mu}^{\,\,\omega}\delta_{\nu}^{\,\,\tau}\\!-\\!\mbox{$\frac{1}{2}$}g^{\tau\omega}g_{\mu\nu}\right)\\!\right)\\!\hat{\Phi}_{\kappa}^{\,\,\nu}\\!+\\!\frac{1}{r}\partial_{\kappa}\varsigma^{(\tau\omega)}\\!(x).\
\ $

As in any standard gauge theory, the field strengths (curvatures) transform
identically as with rigid transformations (not space–time point dependent
parameters). Calling $(\tilde{F}_{\ \varepsilon\kappa}^{{\cal O}})^{\prime}$
the transformed field strength, we have for instance:

$\displaystyle({\tilde{F}}_{\varepsilon\kappa})^{\prime}$ $\displaystyle=$
$\displaystyle{\tilde{F}}_{\varepsilon\kappa}+{\,{\rm
i}\,}\frac{\hat{\varsigma}^{\omega}(x)}{\ell}\left(\\!-\frac{12\epsilon\ell^{2}}{5h}g_{\omega\sigma}\\!\right)\\!{\tilde{F}}_{\
\varepsilon\kappa}^{\sigma}$ (59) $\displaystyle\hskip 8.5359pt+{\,{\rm
i}\,}\frac{\varsigma^{\omega}(x)}{\ell}\left(\\!+\frac{12\epsilon\ell^{2}}{5h}g_{\omega\sigma}\\!\right)\\!\tilde{{\hat{F}}}_{\
\varepsilon\kappa}^{\sigma},$ $\displaystyle(\tilde{{\hat{F}}}_{\
\varepsilon\kappa}^{\rho})^{\prime}$ $\displaystyle=$
$\displaystyle\tilde{{\hat{F}}}_{\ \varepsilon\kappa}^{\rho}+{\,{\rm
i}\,}\frac{\varsigma(x)}{h}(-h)\tilde{{\hat{F}}}_{\
\varepsilon\kappa}^{\rho}+{\,{\rm
i}\,}\frac{\hat{\varsigma}(x)}{\hat{h}}(-\hat{h})\tilde{{\hat{F}}}_{\
\varepsilon\kappa}^{\rho}$ (60) $\displaystyle+$
$\displaystyle\\!\\!\\!\\!{\,{\rm
i}\,}\frac{\hat{\varsigma}^{\sigma}(x)}{\ell}\\!\left(\\!h\delta_{\sigma}^{\,\,\rho}{\tilde{F}}_{\varepsilon\kappa}\\!+\hat{h}\delta_{\sigma}^{\,\,\rho}\tilde{\hat{F}}_{\varepsilon\kappa}\\!-\\!{\,{\rm
i}\,}r^{\prime}\sigma[\tau\omega]_{\sigma}^{\,\,\rho}{\tilde{G}}_{\ \ \
\varepsilon\kappa}^{[\tau\omega]}+r\sigma(\tau\omega)_{\sigma}^{\,\,\rho}{\tilde{H}}_{\
\ \ \varepsilon\kappa}^{(\tau\omega)}\\!\right)\ $ $\displaystyle+$
$\displaystyle\\!\\!\\!\\!{\,{\rm
i}\,}\frac{\varsigma^{[\tau\omega]}(x)}{r^{\prime}}(+{\,{\rm
i}\,}r^{\prime}\sigma[\tau\omega]_{\sigma}^{\,\,\rho})\tilde{{\hat{F}}}_{\
\varepsilon\kappa}^{\sigma}+{\,{\rm
i}\,}\frac{\varsigma^{(\tau\omega)}(x)}{r}(-r\sigma(\tau\omega)_{\sigma}^{\,\,\rho})\tilde{{\hat{F}}}_{\
\varepsilon\kappa}^{\sigma},$

and similar expressions for the other field strengths. The action of
generators on the field strength goes along the corresponding expressions for
the fields (interchanging
$B_{\varepsilon},\hat{B}_{\varepsilon},\cdots,\hat{\phi}_{\varepsilon}^{\,\,\rho}$
respectively with
$\tilde{F}_{\varepsilon\kappa},\tilde{\hat{F}}_{\varepsilon\kappa},\cdots,\tilde{\hat{F}}_{\
\varepsilon\kappa}^{\rho}$ ). For instance,

$\displaystyle\,[T_{\mu},\ell\,\tilde{{\hat{F}}}_{\
\varepsilon\kappa}^{\tau}]_{1}\\!\\!\\!\\!$ $\displaystyle=\\!$
$\displaystyle\\!\\!h\delta_{\mu}^{\,\,\tau}{\tilde{F}}_{\varepsilon\kappa}+\hat{h}\delta_{\mu}^{\,\,\tau}\tilde{{\hat{F}}}_{\varepsilon\kappa}-\\!{\,{\rm
i}\,}r^{\prime}\sigma[\rho\sigma]_{\mu}^{\,\,\tau}{\tilde{G}}_{\ \ \
\varepsilon\kappa}^{[\rho\sigma]}+r\sigma(\rho\sigma)_{\mu}^{\,\,\tau}\\!{\tilde{H}}_{\
\ \ \varepsilon\kappa}^{(\rho\sigma)}\\!\\!\\!,\ \ $
$\displaystyle\,[\bar{T}_{\mu},\ell\,{\tilde{F}}_{\
\varepsilon\kappa}^{\tau}]_{1}\\!\\!\\!\\!$ $\displaystyle=\\!\\!$
$\displaystyle\\!\\!\\!\\!-h\delta_{\mu}^{\,\,\tau}{\tilde{F}}_{\varepsilon\kappa}-\hat{h}\delta_{\mu}^{\,\,\tau}\tilde{\hat{F}}_{\varepsilon\kappa}-\\!{\,{\rm
i}\,}r^{\prime}\sigma[\rho\sigma]_{\mu}^{\,\,\tau}{\tilde{G}}_{\ \ \
\varepsilon\kappa}^{[\rho\sigma]}-r\sigma(\rho\sigma)_{\mu}^{\,\,\tau}\\!{\tilde{H}}_{\
\ \ \varepsilon\kappa}^{(\rho\sigma)}.\ \ \ $ (61)

Now, we would like to construct an invariant Lagrangian associated to the
considered field strengths (curvatures). We use the invariant quadratic
casimir $I$ in (2) as our prime tool. This invariant can be rewritten as
follows:

$\displaystyle I=\left[\frac{{T}^{\omega}}{\ell}\
\frac{\hat{T}^{\omega}}{\ell}\ \frac{M^{\rho\tau}}{r^{\prime}}\
\frac{T^{\rho\tau}}{r}\ \frac{{T_{[o]}}}{h}\
\frac{{\hat{T}_{[o]}}}{\hat{h}}\right]\\!\cdot\\!{\bf
M}\\!\cdot\\!\left[\frac{\hat{T}_{\omega}}{\ell}\ \frac{{T}_{\omega}}{\ell}\
\frac{M_{\rho\tau}}{r^{\prime}}\ \frac{T_{\rho\tau}}{r}\ \frac{{T_{[o]}}}{h}\
\frac{{\hat{T}_{[o]}}}{\hat{h}}\right]^{tr}\\!\\!\\!,\ \ $ (62)
$\displaystyle{\rm with}\ \ {\bf M}:={\rm diag}\left(\frac{2}{\epsilon},\
\frac{2}{\epsilon},\ \frac{1}{1},\ \frac{1}{1},\ \frac{24}{5},\
\frac{1}{5}\right).$ (63)

We consider the field strength dual counterpart to build a lagrangian ${\cal
L}_{o}$. It is easy to verify the invariance of the Lagrangian ${\cal L}_{o}$
under the considered local transformations. For instance $[T_{\mu},{\cal
L}_{o}]=0$. This is equivalent to taking the trace of the matrix products in
the adjoint representation:

$\displaystyle 4{\cal L}_{o}$ $\displaystyle=$ $\displaystyle{\,{\rm
i}\,}[\ell\tilde{{\hat{F}}}^{\ \varepsilon\kappa}_{\omega}\ \ell{\tilde{F}}^{\
\varepsilon\kappa}_{\omega}\ r^{\prime}{\tilde{G}}^{\ \ \
\varepsilon\kappa}_{[\rho\tau]}\ r{\tilde{H}}^{\ \ \
\varepsilon\kappa}_{(\rho\tau)}\ h{\tilde{F}}^{\varepsilon\kappa}\
\hat{h}\tilde{{\hat{F}}}^{\varepsilon\kappa}]$ (65) $\displaystyle\cdot{\bf
M}^{-1}\cdot{\,{\rm i}\,}[\ell{\tilde{F}}_{\ \varepsilon\kappa}^{\omega}\
\ell\tilde{{\hat{F}}}_{\ \varepsilon\kappa}^{\omega}\ r^{\prime}{\tilde{G}}_{\
\ \ \varepsilon\kappa}^{[\rho\tau]}\ r{\tilde{H}}_{\ \ \
\varepsilon\kappa}^{(\rho\tau)}\ h{\tilde{F}}_{\varepsilon\kappa}\
\hat{h}\tilde{{\hat{F}}}_{\varepsilon\kappa}]^{tr}\\!\\!\ \ $ $\displaystyle=$
$\displaystyle-\frac{\epsilon\ell^{2}}{2}\\{\tilde{{\hat{F}}}^{\
\varepsilon\kappa}_{\omega},{\tilde{F}}_{\
\varepsilon\kappa}^{\omega}\\}-{r^{\prime 2}}{\tilde{G}}^{\ \ \
\varepsilon\kappa}_{[\rho\tau]}{\tilde{G}}_{\ \ \
\varepsilon\kappa}^{[\rho\tau]}-{r^{2}}{\tilde{H}}^{\ \ \
\varepsilon\kappa}_{(\rho\tau)}{\tilde{H}}_{\ \ \
\varepsilon\kappa}^{(\rho\tau)}$
$\displaystyle-\frac{5h^{2}}{24}{\tilde{F}}^{\varepsilon\kappa}{\tilde{F}}_{\varepsilon\kappa}-{5\hat{h}^{2}}\tilde{{\hat{F}}}^{\varepsilon\kappa}\tilde{\hat{F}}_{\varepsilon\kappa}.$

We ponder briefly the possibility of expressing the gauge field
$\Phi_{\varepsilon}^{\,\,\sigma}$ and $\bar{\Phi}_{\varepsilon}^{\,\,\sigma}$
(comming from the representation
$(\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$})\otimes(\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$})$
of the Lorentz–index vector label of the generator $T_{\sigma}$ or
$\bar{T}_{\sigma}$ and the Einstein-index vector label of the covariant
derivative) in terms of its trace through a complex scalar field $\phi$ and
some further contributions (probably in the form of an antisymmetric vector or
a spin $(1,1)$ symmetric tensor), setting all other gauge fields to zero. We
obtain:

$\displaystyle\Phi_{\varepsilon}^{\,\,\sigma}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{\delta_{\varepsilon}^{\,\,\sigma}\sqrt{2}}{\sqrt{3|\epsilon|}\ell}\phi+\cdots,$
(66) $\displaystyle\\!\\!\\!{\cal L}_{o}|_{restr}\\!\\!\\!$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle\\!\\!\\!-\frac{\epsilon\ell^{2}}{8}\\{{\tilde{\hat{F}}}^{\
\varepsilon\kappa}_{\omega}\\!,{\tilde{F}}_{\
\varepsilon\kappa}^{\omega}\\}-\\!{\frac{r^{\prime 2}}{4}}{\tilde{G}}^{\ \ \
\varepsilon\kappa}_{[\rho\tau]}{\tilde{G}}_{\ \ \
\varepsilon\kappa}^{[\rho\tau]}\\!=\\!-\frac{\epsilon}{|\epsilon|}\partial_{\varepsilon}\\!\phi\partial^{\varepsilon}\bar{\phi}-\\!\frac{8}{3}{(\phi\bar{\phi})}^{2}\\!\\!+\\!\cdots\\!.\
\ \ $ (67)

This would require $\epsilon$ to be negative. An extra quadratic soft breaking
mass term might complete a Higgs-like potential. The quartic self interaction
of the would-be Higgs field comes from ${\tilde{G}}_{\ \
\varepsilon\kappa}^{[\rho\tau]}$, responsible for local Lorentz transformation
invariance, although with a huge coupling constant. The corresponding versions
of (33) and (38) would embody the aimed mixture between gauge and Higgs field.
Nevertheless, although this field might contribute to the Higgs field perhaps
after some symmetry breaking, there are fields that will properly embody the
Higgs as we will see in a Poincaré–like extension [7]. Beware: in using
$\delta_{\varepsilon}^{\,\,\sigma}$ in (66) we identify an Einstein
(non–Lorentz transforming) index $\varepsilon$ with a Lorentz transforming
index $\sigma$. In fact, the transformations (33-34) and (38) represent rather
a mixing between the gauge fields $hB_{b}$, $\hat{h}\hat{B}_{b}$ and the
tetrad–like fields $\ell\Phi_{b}^{\,\tau}$, $\ell\hat{\Phi}_{b}^{\,\tau}$. The
transformation between Higgs–like fields demand either a wider realm of
covariant derivatives (explored below), or a Poincaré–like extension.

The kinetic parts of the gauge fields $hB_{b}$, $\hat{h}\hat{B}_{b}$ will give
the proper sign for kinetic terms ( where “tdt.” stands for total derivative
terms)

$\displaystyle-\frac{5h^{2}}{24\cdot
4}(\partial_{\varepsilon}B_{\kappa}\\!-\partial_{\kappa}B_{\varepsilon})(\partial^{\varepsilon}B^{\kappa}\\!-\partial^{\kappa}B^{\varepsilon})=\frac{5h^{2}}{24\cdot
4}(2B_{\kappa}(\partial_{\varepsilon}\partial^{\varepsilon}B^{\kappa})\\!+2(\partial_{\rho}B^{\rho})^{2})+{\rm
tdt.},\ \ $
$\displaystyle-\frac{5\hat{h}^{2}}{4}(\partial_{\varepsilon}\hat{B}_{\kappa}\\!-\partial_{\kappa}\hat{B}_{\varepsilon})(\partial^{\varepsilon}\hat{B}^{\kappa}\\!-\partial^{\kappa}\hat{B}^{\varepsilon})=\frac{5\hat{h}^{2}}{4}(2\hat{B}_{\kappa}(\partial_{\varepsilon}\partial^{\varepsilon}\hat{B}^{\kappa})\\!+2(\partial_{\rho}\hat{B}^{\rho})^{2})+{\rm
tdt.}.\ \ $

We could naively try to reduce the complexity of kinetic terms of other gauge
fields carrying multiple Lorentz indices. We explore if the real massless
fields ${B}_{\varepsilon}^{\ (\rho\tau)}$ and ${B}_{\varepsilon}^{\
[\rho\tau]}$ can be cast in terms of real symmetric vector $S_{\rho}$,
$U_{\rho}$, and $V_{\rho}$:

$\displaystyle{B}_{\varepsilon}^{\ (\rho\sigma)}$
$\displaystyle=\delta_{\varepsilon}^{\rho}S^{\sigma}+\delta_{\varepsilon}^{\sigma}S^{\rho}-\mbox{$\frac{1}{2}$}g^{\rho\sigma}S_{\varepsilon},\hskip
71.13188pt$ (68) $\displaystyle{B}_{\varepsilon}^{\ [\rho\sigma]}$
$\displaystyle=\tilde{b}(\delta_{\varepsilon}^{\rho}U^{\sigma}-\delta_{\varepsilon}^{\sigma}U^{\rho})+\tilde{a}\epsilon^{\tau\omega\rho\sigma}(g_{\varepsilon\tau}V_{\omega}-g_{\varepsilon\omega}V_{\tau}),$
(69)

using $\epsilon^{0123}=-\epsilon^{0123}=1$. This would induce kinetic parts
given by

$\displaystyle\\!\\!-\frac{r^{2}}{4}(\partial_{\varepsilon}B_{\kappa}^{\,\,(\tau\omega)}\\!-\partial_{\kappa}B_{\varepsilon}^{\,\,(\tau\omega)})(\partial^{\varepsilon}B^{\kappa}_{\,\,(\tau\omega)}\\!-\partial^{\kappa}B^{\varepsilon}_{\,\,(\tau\omega)})$
$\displaystyle\ \
={r^{2}}(4S_{\kappa}(\partial_{\varepsilon}\partial^{\varepsilon}S^{\kappa})\\!+2(\partial_{\rho}S^{\rho})^{2})+{\rm
tdt.},$ $\displaystyle\\!\\!\\!\\!-\frac{r^{\prime
2}}{4}(\partial_{\varepsilon}B_{\kappa}^{\,\,[\tau\omega]}\\!-\partial_{\kappa}B_{\varepsilon}^{\,\,[\tau\omega]})(\partial^{\varepsilon}B^{\kappa}_{\,\,[\tau\omega]}\\!-\partial^{\kappa}B^{\varepsilon}_{\,\,[\tau\omega]})\\!$
$\displaystyle\\!=\\!{r^{\prime
2}\tilde{b}^{2}}(2U_{\kappa}(\partial_{\varepsilon}\partial^{\varepsilon}U^{\kappa}\\!)\\!-\\!(\partial_{\rho}U^{\rho})^{2})\\!+\\!{r^{\prime
2}\tilde{a}^{2}}$
$\displaystyle(\\!-8V_{\kappa}(\partial_{\varepsilon}\partial^{\varepsilon}V^{\kappa}\\!)\\!+4(\partial_{\rho}V^{\rho})^{2})\\!+\\!{\rm
tdt.}\ \ \ \ $

Such naive trials seem to lead to no ghosts for the choice
$\tilde{b}>0,\tilde{a}=0$, if we maintain the contribution of longitudinal
parts $(\partial_{\rho}U^{\rho})^{2}$ under control, which is done by a proper
gauge fixing and Faddeev-Popov or BRST formalism.

##  4 Differential Lorentz Extension Representation

We explore the representability of the Lorentz Extension LE in terms of
differential operators in a geometric space including the 4D space-time
coordinates. To each symmetry generator ${\cal O}$ we assign a differential
operator $-{\,{\rm i}\,}\delta_{{\cal O}}$ that when acting on the fields
represents the same algebra. If $\lambda^{{\cal O}^{\prime}}{\cal
C}_{a}^{{\cal O}^{\prime}}\\!(x)$ is a function of the geometric–frame
coordinates (such as the representation on the fields i.e. left Lie–algebra
module), then, besides the transformation considered in (33)–(52), we
introduce a differential operator representation $-{\,{\rm i}\,}\delta_{{\cal
O}}$ which copes with the dependence on the geometric frame

$\displaystyle\,[{\cal O},\lambda^{{\cal O}^{\prime}}{\cal C}_{a}^{{\cal
O}^{\prime}}]_{2}$ $\displaystyle=-{\,{\rm i}\,}\delta_{{\cal
O}}(\lambda^{{\cal O}^{\prime}}{\cal C}_{a}^{{\cal O}^{\prime}}),$
$\displaystyle\ [{\cal O},\lambda^{{\cal O}^{\prime}}{\cal C}_{a}^{{\cal
O}^{\prime}}]_{1}$ $\displaystyle=-{\,{\rm i}\,}\,C_{\ {\cal O}{\cal
O}^{\prime\prime}}^{{\cal O}^{\prime}}\lambda^{{\cal O}^{\prime\prime}}{\cal
C}_{a}^{{\cal O}^{\prime\prime}},$ (70)

with $\lambda^{{\cal O}}=\lambda_{{\cal O}}$ the coupling associated to
${{\cal O}}$.The indices in the coupling constants do not trigger summations
over repeated indices. Since we want a differential representation which
commutes with the action of the abstract symmetry generators ${\cal O}$, it
turns out that the algebra of differential operators should satisfy

$\,[{\,{\rm i}\,}\delta_{{\cal O}},{\,{\rm i}\,}\delta_{{\cal
O}^{\prime}}]={\,{\rm i}\,}C^{{\cal O}^{\prime\prime}}_{{\cal O}{\cal
O}^{\prime}}({\,{\rm i}\,}\delta_{{\cal O}^{\prime\prime}}),\ {\rm for}\
[{\cal O},{\cal O}^{\prime}]={\,{\rm i}\,}C^{{\cal O}^{\prime\prime}}_{{\cal
O}{\cal O}^{\prime}}{{\cal O}^{\prime\prime}}.$ (71)

There are (at least) two ways of representing the algebra (5)–(15) as
differential operators. The diverse generators can either be represented in
terms of Hermitian (H) and Anti–Hermitian (AH) operators, or in terms of
Hermitian (H) and Adjoint–Pairs (AP) of generators. The nature of generators
might change under symmetry breaking. Here is a description of the found
differential representations:

${\footnotesize\begin{array}[]{c|cccccc}\hline\cr{\stackrel{{\scriptstyle{\rm
Generator}\rightarrow}}{{\downarrow{\rm Diff.Rep.}}}}&\ \ \ \ {T}_{\rho}&\ \ \
\ \hat{T}_{\rho}&\ \ \ \ M_{\rho\tau}&\ \ \ \ T_{\rho\tau}&\ \ \ \ {T_{[o]}}&\
\ \ \ {\hat{T}_{[o]}}\\\ \hline\cr 5D&{\rm H}&{\rm H}&{\rm H}&{\rm AH}&{\rm
AH}&{\rm AH}\\\ \hline\cr 10D&{\rm AP}&{\rm AP}&{\rm H}&{\rm H}&{\rm H}&{\rm
H}\\\ \hline\cr\end{array}}$

Table 1: Differential Representations; $\rho,\tau=0,1,2,3.$

We first present the Brane or 5D representation in terms of differential
operators: We introduce besides the 4D space–time coordinates $\chi_{\rho}$
with $\partial_{\rho}=\partial/\partial\chi^{\rho},\ \rho=0,1,2,3$, a new
variable $\chi_{5}$, with $\partial_{5}=\partial/\partial\chi^{5}$, with
$\partial_{5}(\chi_{5})=\epsilon=g_{55}$. This variable will be associated to
the symmetry combining Higgs and gauge fields. We allow for another
independent variable $\chi_{4}$, with
$\partial_{4}=\partial/\partial\chi^{4}$, with
$\partial_{4}(\chi_{4})=g_{44}$. This is the variable that will lead to an
extension of the Poincaré algebra after an optional contraction. Since the
coordinate $\chi_{4}$ might be removed after the contraction procedure, we do
not count this coordinate in the total dimension of the representation. We
represent $M_{\mu\nu}$ by $-{\,{\rm
i}\,}(\chi_{\mu}\partial_{\nu}-\chi_{\nu}\partial_{\mu})$, and $T_{\mu\nu}$ by
the Anti-Hermitic operator
$+(\chi_{\mu}\partial_{\nu}+\chi_{\nu}\partial_{\mu}-\mbox{$\frac{1}{2}$}g_{\mu\nu}\chi^{\rho}\partial_{\rho})$.
We adopt $-{\,{\rm i}\,}\delta_{T_{\mu}}=+\sqrt{2}{\,{\rm
i}\,}\chi_{5}\partial_{\mu},\ -{\,{\rm
i}\,}\delta_{\hat{T}_{\nu}}=-\sqrt{2}{\,{\rm i}\,}\chi_{\nu}\partial_{5}$ to
obtain the 5D representation:

$\displaystyle-{\,{\rm i}\,}\delta_{M_{\mu\nu}}$ $\displaystyle=$
$\displaystyle\\!\\!-{\,{\rm
i}\,}(\chi_{\mu}\partial_{\nu}-\chi_{\nu}\partial_{\mu}),\ -{\,{\rm
i}\,}\delta_{T_{\mu\nu}}\\!=+(\chi_{\mu}\partial_{\nu}\\!+\\!\chi_{\nu}\partial_{\mu}\\!-\\!\mbox{$\frac{1}{2}$}g_{\mu\nu}\chi^{\rho}\partial_{\rho}),\
\ $ (72) $\displaystyle-{\,{\rm i}\,}\delta_{T_{\mu}}$ $\displaystyle=$
$\displaystyle\\!\\!+\sqrt{2}{\,{\rm i}\,}\chi_{5}\partial_{\mu},\ \ -{\,{\rm
i}\,}\delta_{\hat{T}_{\nu}}=-\sqrt{2}{\,{\rm i}\,}\chi_{\nu}\partial_{5},$
(73) $\displaystyle-{\,{\rm i}\,}\delta_{{T_{[o]}}}$ $\displaystyle=$
$\displaystyle\\!\\!-\frac{5}{6}\epsilon^{-1}\chi_{5}\partial_{5}+\frac{1}{6}g_{44}^{-1}\chi_{4}\partial_{4}+\frac{1}{6}\chi^{\rho}\partial_{\rho},$
(74) $\displaystyle-{\,{\rm i}\,}\delta_{{\hat{T}_{[o]}}}$ $\displaystyle=$
$\displaystyle-4g_{44}^{-1}\chi_{4}\partial_{4}+\chi^{\rho}\partial_{\rho}.$
(75)

The quantity
$|\chi|^{2}=\chi^{\rho}\chi_{\rho}+g_{44}\chi^{4}\chi^{4}+\epsilon\chi^{5}\chi^{5}$,
where $\chi^{5}=\epsilon^{-1}\chi_{5}$, and $\chi^{4}=g_{44}^{-1}\chi_{4}$, is
not invariant under the Anti–Hermitic differential operators, precluding a
simple geometric space interpretation with metric ${\rm
diag(1,-1,-1,-1,g_{44},\epsilon)}$ for the LE, but it is invariant under the
subalgebra $<\\{M_{\mu\nu},T_{\rho}+\hat{T}_{\rho}\\}>$. Nevertheless, the
quantity ${\cal
I}=(\chi^{\mu}\chi_{\mu})^{4}(\chi^{4}\chi_{4})(\chi^{5}\chi_{5})$, is
invariant under $\delta_{M_{\mu\nu}},\delta_{{T_{[o]}}}$, and
$\delta_{{\hat{T}_{[o]}}}$. Now, ${\cal I}$ is invariant under $-{\,{\rm
i}\,}\bar{\xi}^{\mu}\delta_{T_{\mu}}-{\,{\rm
i}\,}\xi^{\mu}\delta_{\hat{T}_{\mu}}$ for $\xi^{\mu}$ real (or $+{\,{\rm
i}\,}\bar{\xi}^{\mu}\delta_{T_{\mu}}-{\,{\rm
i}\,}\xi^{\mu}\delta_{\hat{T}_{\mu}}$ for $\xi^{\mu}$ imaginary) and
$\chi^{5}\chi_{5}=(1/4)\chi^{\mu}\chi_{\mu}$, which looks like an on–shell or
perhaps brane condition. For such choices $[-{\,{\rm
i}\,}\bar{\xi}^{\mu}\delta_{T_{\mu}}-{\,{\rm
i}\,}\xi^{\mu}\delta_{\hat{T}_{\mu}},-{\,{\rm
i}\,}\bar{\kappa}^{\mu}\delta_{T_{\mu}}-{\,{\rm
i}\,}\kappa^{\mu}\delta_{\hat{T}_{\mu}}]$ has only nonvanishing contributions
proportional to $-{\,{\rm i}\,}\delta_{M_{\mu\nu}}$, which retains invariance.
We do not verify invariance of $|\chi|^{2}$ under
$\xi^{(\mu\nu)}\delta_{T_{\mu\nu}}$ for arbitrary parameter $\xi^{(\mu\nu)}$.
So, with this tenets, only a subgroup of the original extension seem to be
cast with this representation. Therefore the label of Brane representation
since it might be only relevant after some symmetry breaking or some
topological reconfiguration into a brane.

The contraction is adjusted to provide a differential representation without
involving the $x^{4}$ coordinate. We use $R\rightarrow\infty$,
$\chi_{4}/R\rightarrow(1/2)$,
$g^{44}\chi_{4}\partial_{4}\rightarrow-\epsilon^{-1}\chi_{5}\partial_{5}-\chi^{\rho}\partial_{\rho}+3$,
and $(1/R)\partial_{4}\rightarrow 0$.

We naively expect that $T_{\mu}$ and $\hat{T}_{\mu}$ (as the Higgs field)
build an adjoint pair, as hinted by (2) and (33), but this is not
substantiated in the 5D representation. There is a way to restore naive
expectations at the expense of complexifying the coordinates. This leads to
the 10D representation. Let $z_{a}=\chi_{a}+{\,{\rm i}\,}y_{a},\
\bar{z}_{a}=\chi_{a}-{\,{\rm i}\,}y_{a}$ for $a=0,1,2,3,4,5$. We find a
representation with

$\displaystyle\bar{z}^{a}z_{a}$ $\displaystyle=$ $\displaystyle
g_{\mu\nu}\bar{z}^{\mu}{z}^{\nu}+g_{44}\bar{z}^{4}{z}^{4}+\epsilon\bar{z}^{5}{z}^{5}$
(76) $\displaystyle=$ $\displaystyle
g_{\mu\nu}{\chi}^{\mu}{\chi}^{\nu}+g_{\mu\nu}{y}^{\mu}{y}^{\nu}+g_{44}{\chi}^{4}{\chi}^{4}+g_{44}{y}^{4}{y}^{4}+\epsilon{\chi}^{5}{\chi}^{5}+\epsilon{y}^{5}{y}^{5}$

invariant, giving substance to a relation to a subalgebra of an
$su(s_{4},t_{4})$ with $s_{4},t_{4}>0$ and $s_{4}+t_{4}=6$, such as
$u(s_{1},t_{1})$ with $s_{1},t_{1}>0$ and $s_{1}+t_{1}=5$. Adopting $\epsilon$
negative seems to assign a space–like role to the $\chi^{5},y^{5}$
coordinates. Let us denote $\partial_{a}=\partial/\partial\chi^{a}$ and
$\hat{\partial}_{a}=\partial/\partial y^{a}$. The 10D representation with such
invariance is:

$\displaystyle-{\,{\rm i}\,}\delta_{M_{\mu\nu}}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!-{\,{\rm
i}\,}(\chi_{\mu}\partial_{\nu}-\chi_{\nu}\partial_{\mu})-{\,{\rm
i}\,}(y_{\mu}\hat{\partial}_{\nu}-y_{\nu}\hat{\partial}_{\mu}),$ (77)
$\displaystyle-{\,{\rm i}\,}\delta_{T_{\mu\nu}}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!-{\,{\rm
i}\,}(\chi_{\mu}\hat{\partial}_{\nu}+\chi_{\nu}\hat{\partial}_{\mu}-\\!\mbox{$\frac{1}{2}$}g_{\mu\nu}\chi^{\rho}\hat{\partial}_{\rho})\\!+\\!{\,{\rm
i}\,}(y_{\mu}\partial_{\nu}+y_{\nu}\partial_{\mu}-\\!\mbox{$\frac{1}{2}$}g_{\mu\nu}y^{\rho}\partial_{\rho}),\
\ $ (78) $\displaystyle-{\,{\rm i}\,}\delta_{T_{\mu}}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!+\frac{{\,{\rm
i}\,}}{\sqrt{2}}\left(\\!(\chi_{5}+\\!{\,{\rm
i}\,}y_{5})(\partial_{\mu}-\\!{\,{\rm
i}\,}\hat{\partial}_{\mu})-(\chi_{\mu}-\\!{\,{\rm
i}\,}y_{\mu})(\partial_{5}+\\!{\,{\rm i}\,}\hat{\partial}_{5})\right)\\!,$
(79) $\displaystyle-{\,{\rm i}\,}\delta_{\hat{T}_{\nu}}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!-\frac{{\,{\rm
i}\,}}{\sqrt{2}}\left(\\!(\chi_{\mu}+\\!{\,{\rm
i}\,}y_{\mu})(\partial_{5}-\\!{\,{\rm
i}\,}\hat{\partial}_{5})-(\chi_{5}-\\!{\,{\rm
i}\,}y_{5})(\partial_{\mu}+\\!{\,{\rm i}\,}\hat{\partial}_{\mu})\right)\\!,$
(80) $\displaystyle-{\,{\rm i}\,}\delta_{{T_{[o]}}}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!-\frac{5{\,{\rm
i}\,}}{6}\epsilon^{\\!-1}(y_{5}\partial_{5}\\!-\\!\chi_{5}\hat{\partial}_{5})\\!+\\!\frac{{\,{\rm
i}\,}}{6}g_{44}^{-1}(y_{4}\partial_{4}\\!-\\!\chi_{4}\hat{\partial}_{4})\\!+\\!\frac{{\,{\rm
i}\,}}{6}(y^{\rho}\partial_{\rho}\\!-\\!\chi^{\rho}\hat{\partial}_{\rho}),\ $
(81) $\displaystyle-{\,{\rm i}\,}\delta_{{\hat{T}_{[o]}}}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!-4{\,{\rm
i}\,}g_{44}^{-1}(y_{4}\partial_{4}-\\!\chi_{4}\hat{\partial}_{4})+{\,{\rm
i}\,}(y^{\rho}\partial_{\rho}-\\!\chi^{\rho}\hat{\partial}_{\rho}).$ (82)

The contraction procedure is adjusted to provide a differential representation
without the involvement of the $\chi^{4},y^{4}$ coordinates. Use
$R\rightarrow\infty$, $\chi_{4}/R\rightarrow 1$,
$g^{44}(y_{4}\partial_{4}-\\!\chi_{4}\hat{\partial}_{4})\rightarrow-\epsilon^{-1}(y_{5}\partial_{5}-\\!\chi_{5}\hat{\partial}_{5})-(y^{\rho}\partial_{\rho}-\\!\chi^{\rho}\hat{\partial}_{\rho})$,
$y_{4}/R\rightarrow 0$, $\partial_{4}/R\rightarrow 0$, and
$\hat{\partial_{4}}/R\rightarrow 0$. The invariance of (76) dropping the
$\chi^{4}$–coordinate is easily verified. Mechanisms for transition (before or
after contraction) from the 10D to 5D representation will be tackled
elsewhere.

We consider a new base $\tilde{N}_{AB}$ for generators of 10D, with complex
derivation $\partial_{Z^{A}}=(1/2)(\partial_{x^{A}}-{\,{\rm
i}\,}\partial_{y^{A}}),\ \partial_{\bar{Z}^{A}}=(1/2)(\partial_{x^{A}}+{\,{\rm
i}\,}\partial_{y^{A}})$, and find:

$\displaystyle-{\,{\rm i}\,}\delta_{\tilde{N}_{AB}}$ $\displaystyle=$
$\displaystyle-{\,{\rm
i}\,}(z_{A}\partial_{z^{B}}-\bar{z}_{B}\partial_{\bar{z}^{A}}),\ \
A,B=0,1,2,3,4,5.$ (83) $\displaystyle\,[\tilde{N}_{AB},\tilde{N}_{CD}]$
$\displaystyle=$ $\displaystyle-{\,{\rm
i}\,}(g_{AD}\tilde{N}_{CB}-g_{BC}\tilde{N}_{AD}).$ (84)

Lowering and rising of labels via ${\rm diag(1,-1,-1,-1,g_{44},\epsilon)}$.
Using (77)–(82) we relate the old and new basis:

$\displaystyle M_{\mu\nu}$ $\displaystyle=$
$\displaystyle\tilde{N}_{\mu\nu}-\tilde{N}_{\nu\mu},$ (85) $\displaystyle
T_{\mu\nu}$ $\displaystyle=$ $\displaystyle{\,{\rm
i}\,}(\tilde{N}_{\mu\nu}+\tilde{N}_{\nu\mu})-\frac{{\,{\rm
i}\,}g_{\mu\nu}}{2}\tilde{N}^{\rho}_{\ \rho},$ (86) $\displaystyle T_{\mu}$
$\displaystyle=$ $\displaystyle-\sqrt{2}\tilde{N}_{5\mu},\ \ \hat{T}_{\mu}\ =\
+\sqrt{2}\tilde{N}_{\mu 5},$ (87) $\displaystyle T_{[o]}$ $\displaystyle=$
$\displaystyle-\frac{5{\,{\rm i}\,}}{6}\tilde{N}^{4}_{\ 4}+\frac{{\,{\rm
i}\,}}{6}\tilde{N}^{5}_{\ 5}+\frac{{\,{\rm i}\,}}{6}\tilde{N}^{\rho}_{\
\rho},$ (88) $\displaystyle\hat{T}_{[o]}$ $\displaystyle=$
$\displaystyle-4{\,{\rm i}\,}\tilde{N}^{4}_{\ 4}+{\,{\rm
i}\,}\tilde{N}^{\rho}_{\ \rho}.$ (89)

Using this basis we express the quadratic casimir (2) in terms of the new
basis:

$\displaystyle\frac{1}{4}\,I$ $\displaystyle=-\tilde{N}^{\mu
5}\tilde{N}_{5\mu}-\tilde{N}^{5\mu}\tilde{N}_{\mu
5}-\tilde{N}^{\mu\nu}\tilde{N}_{\nu\mu}-\tilde{N}^{55}\tilde{N}_{55}-\tilde{N}^{44}\tilde{N}_{44}$
(90) $\displaystyle\ \ +\frac{1}{6}(\tilde{N}^{\mu}_{\ \mu}+\tilde{N}^{4}_{\
4}+\tilde{N}^{5}_{\ 5})^{2}.$

Similar elaborations can be made after contraction.

The set of generators $\tilde{N}_{AB}$ involved in the LE satisfy an
$u(t_{1},s_{1})$ algebra with $t_{1}+s_{1}=5$, and $t_{1}=1$ for
$\epsilon=-1$. The central charge of the LE $-5{\cal T}=T_{[o]}-\hat{T}_{[o]}$
$={\,{\rm i}\,}(4\tilde{N}^{4}_{\ 4}+\tilde{N}^{5}_{\ 5})-(5/6){\,{\rm
i}\,}{\bf D}$, with ${\bf D}$ the complex 6D trace
$\sum_{A\in\\{\mu,4,5\\}}\tilde{N}^{A}_{\ A}=\tilde{N}^{\mu}_{\
\mu}+\tilde{N}^{4}_{\ 4}+\tilde{N}^{5}_{\ 5}$ of novel generators in (83–89).
The obtained LE quadratic casimir $I$ in (2) and (90) lead to congecture a
quartic casimir $\check{I}$ of the following form after contraction:

$\displaystyle\frac{1}{2^{4}}\check{I}=-\\!\\!\\!\\!\\!\sum_{A,B,C,D\in\\{\mu,5\\}}\\!\\!\\!\\!\\!\tilde{N}^{AB}\\!\tilde{N}_{BC}\tilde{N}^{CD}\tilde{N}_{DA}.$
(91)

In a prospective Poincaré extension, the quadratic casimir will be a summation
of squares of our adopted generators. Our choices have not been directed to
suit just the Lorentz extension. A Poincaré extension will have generous
possibilities for devising mechanisms affording a transition to a stage
corresponding to the standard model prior internal gauge symmetry breaking.

##  5 Fields and a Locally Invariant Model II

We address now the use of differential representations for having actual
external symmetries transforming also the geometric frame (extended
space–time). We notice that the operator $\partial_{\mu}$ usually inserted in
the covariant derivative do not close an algebra with the obtained
differential operators $-{\,{\rm i}\,}\,\delta_{{\cal O}}$. Nevertheless, we
observe that the operator $-{\,{\rm i}\,}\delta_{T_{\mu}}$ in the 5D
representation has a close resemblance to the one used to represent
translations. It is not ruled out that through some mechanism, it might
contribute to the Standard model translations. In this model, we could perhaps
abandon the variable $\chi_{4}$ all together since there is already a copy of
the Poincaré algebra lurking in the obtained extension. We do not know if both
Higgs and gravitational interaction could be recovered from such a model, but
it is worth pursuing this line of thought for a moment. When introducing
further internal symmetries, the $T_{\mu}$ becomes $T_{\alpha\mu}$ with
$\alpha$ a fundamental $SU(2)$ representation label associated to a complex
coordinate. So, the variable $\chi_{4}$ for getting separately translation
generators will be required.

We use either of the differential representations of the previous section to
insert the action of the symmetry on the geometric frame with coordinates
either $(\chi_{0},\chi_{1},\chi_{2},\chi_{3},\chi_{4},\chi_{5})$, or
$(z_{0},z_{1},z_{2},z_{3},z_{4},z_{5})$ and the fields are functions thereof.
We are forced to depart from the standard gauge field formalism.

We propose finally as covariant derivative a naive extrapolation of the gauge
formalism. In gauge theories, the covariant derivatives are deformations of
the translation operators, which actually belong to a differential
representation of the Poincaré algebra. So, we construct a gauge theory over
the Lie group manifold itself, in the expectation that further contraction or
symmetry breaking procedures will land in the 3+1 dimensional Minkowski space.
The fields and field strengths have now both naive energy dimension 0 will
resemble representation metric array and structure constants respectively:

$\displaystyle{D}_{{\cal O}}\\!\\!\\!\\!$ $\displaystyle:=$
$\displaystyle{\,{\rm i}\,}(-{\,{\rm i}\,}\delta_{{\cal O}})\otimes
Id+\sum_{\tilde{\cal O}}{\,{\rm i}\,}\lambda^{\tilde{\cal O}}{\cal A}_{{\cal
O}}^{\ \tilde{\cal O}}\otimes{\tilde{\cal O}},$ (92)
$\displaystyle\sum_{\tilde{\cal O}}{\,{\rm i}\,}\lambda^{\tilde{\cal O}}{\bf
F}_{\ {\cal O}{\cal O}^{\prime}}^{\tilde{\cal O}}\otimes{\tilde{\cal
O}}\\!\\!$ $\displaystyle:=$ $\displaystyle\,[{D}_{{\cal O}},{D}_{{\cal
O}^{\prime}}]+{\,{\rm i}\,}{D}_{[{\cal O},{\cal O}^{\prime}]},$ (93)
$\displaystyle\lambda^{\hat{\cal O}}{\bf F}_{\ {\cal O}{\cal
O}^{\prime}}^{\hat{\cal O}}:=\delta_{{\cal O}}(\lambda^{\hat{\cal O}}{\cal
A}_{{\cal O}^{\prime}}^{\ \hat{\cal O}})-\delta_{{\cal
O}^{\prime}}(\lambda^{\hat{\cal O}}{\cal A}_{{\cal O}}^{\ \hat{\cal
O}})+\lambda^{\hat{\cal O}}{\bf{\cal F}}_{\ {\cal O}{\cal
O}^{\prime}}^{\hat{\cal O}}+{\,{\rm i}\,}({\,{\rm i}\,}C_{\ {\cal O}{\cal
O}^{\prime}}^{{\cal O}^{\prime\prime}})(\lambda^{\hat{\cal O}}{\cal A}_{{\cal
O}^{\prime\prime}}^{\ \hat{\cal O}}),$ (94) $\displaystyle\lambda^{\hat{\cal
O}}{\bf{\cal F}}_{\ {\cal O}{\cal O}^{\prime}}^{\hat{\cal O}}:={-{\,{\rm
i}\,}}(\lambda^{\tilde{\cal O}}{\cal A}_{{\cal O}}^{\ \tilde{\cal
O}})[\tilde{{\cal O}},\lambda^{\hat{\cal O}}{\cal A}_{{\cal O}^{\prime}}^{\
\hat{\cal O}}(x)]_{1}={{\,{\rm i}\,}}(\lambda^{\tilde{\cal O}}{\cal A}_{{\cal
O}}^{\ \tilde{\cal O}}){\,{\rm i}\,}C_{\ \tilde{\cal O}{\cal
O}^{\prime\prime}}^{\hat{\cal O}}(\lambda^{{\cal O}^{\prime\prime}}{\cal
A}_{{\cal O}^{\prime}}^{\ {\cal O}^{\prime\prime}}).\ \ $ (95)

The calligraphic–noted term $\lambda^{\hat{\cal O}}{\bf{\cal F}}_{\ {\cal
O}{\cal O}^{\prime}}^{\hat{\cal O}}$ is again quadratic in the fields, and
coincides with the expressions in (25–30) changing the indices
$\varepsilon,\kappa$ by generator labels ${\cal O},{\cal O}^{\prime}$. Indices
${\cal O},{\cal O}^{\prime},$ etc. run over the generators of the Lorentz
extension, and the differential operators $-{\,{\rm i}\,}\delta_{{\cal O}}$
are those of the previous section. Later on one can consider the restriction
to covariant derivatives ${D}_{{\cal O}}$ for ${\cal O}$ generators of an
invariant subalgebra. We maintain Einstein summation convention over upper and
lower repeated indices (except for those in coupling constants). The
$\lambda^{\tilde{\cal O}}$, ${\cal A}_{{\cal O}}^{\ \tilde{\cal O}}(x)$, ${\bf
F}_{\ {\cal O}{\cal O}^{\prime}}^{\tilde{\cal O}}(x)$ are respectively the
coupling constant, field, and field strength, and $(x)$ stands for coordinates
in the adopted differential representation. Both field, and field strength are
left–modules over the extended Lorentz Lie algebra, transforming only with
respect to the upper index:

$\displaystyle\ [\tilde{{\cal O}},\lambda^{\hat{\cal O}}{\cal A}_{{\cal O}}^{\
\hat{\cal O}}(x)]_{1}$ $\displaystyle=-{\,{\rm i}\,}\,C_{\ \tilde{{\cal
O}}{\cal O}^{\prime\prime}}^{\hat{{\cal O}}}\lambda^{{\cal
O}^{\prime\prime}}{\cal A}_{{\cal O}}^{\ {\cal O}^{\prime\prime}}(x),$
$\displaystyle\ [\tilde{{\cal O}},\lambda^{\hat{\cal O}}{\bf F}_{\ {\cal
O}{\cal O}^{\prime}}^{\hat{\cal O}}(x)]_{1}$ $\displaystyle=-{\,{\rm
i}\,}\,C_{\ \tilde{{\cal O}}{\cal O}^{\prime\prime}}^{\hat{\cal
O}}\lambda^{{\cal O}^{\prime\prime}}{\bf F}_{\ {\cal O}{\cal
O}^{\prime}}^{{\cal O}^{\prime\prime}}(x).$ (96)

Using the matrix notation ($Id$ being the identity matrix),

$\displaystyle\,[D_{\cal O}]_{\hat{\cal O}^{\prime}}^{\ \ \hat{\cal
O}}=\delta_{\cal O}[Id]_{\hat{\cal O}^{\prime}}^{\ \ \hat{\cal O}}+{\,{\rm
i}\,}(\lambda^{\tilde{\cal O}}{\cal A}_{\cal O}^{\ \tilde{\cal
O}})[R_{\tilde{\cal O}}]_{\hat{\cal O}^{\prime}}^{\ \ \hat{\cal O}},$ (97)
$\displaystyle{\rm with}\ \ [R_{\cal O}]_{\hat{\cal O}^{\prime}}^{\ \
\hat{\cal O}}=-{\,{\rm i}\,}C_{\ {\cal O}\hat{\cal O}^{\prime}}^{\hat{\cal
O}},$ (98)

we can recover the results in (93–94). $R_{\cal O}$ is the well known adjoint
representation.

We generalize the Ansatz for first order gauge transformations with
$(x)$–dependent parameters $\varsigma(x),\hat{\varsigma}(x)$,
$\hat{\varsigma}^{\nu}(x),\varsigma^{\nu}(x)$,
$\varsigma^{[\tau\kappa]}(x),\varsigma^{(\tau\kappa)}(x)$ associated
respectively to generators ${T_{[o]}},\hat{T}_{[o]},T_{\nu}$,
$\hat{T}_{\nu},M_{\tau\kappa}$, $T_{\tau\kappa}$:

$\displaystyle\\!\\!(\lambda^{\hat{{\cal O}}}{{\cal A}_{{\cal O}}^{\ \hat{\cal
O}}})\\!\mapsto\\!(\lambda^{\hat{{\cal O}}}{{\cal A}_{{\cal O}}^{\ \hat{\cal
O}}})^{\prime}\\!\\!=\\!(\lambda^{\hat{{\cal O}}}{{\cal A}_{{\cal O}}^{\
\hat{\cal O}}})\\!+\\!\sum_{\tilde{{\cal O}}}\\!{\,{\rm
i}\,}{\varsigma}^{\tilde{{\cal O}}}(x)[\tilde{{\cal O}},\lambda^{\hat{\cal
O}}{\cal A}_{{\cal O}}^{\ \hat{{\cal O}}}]_{1}\\!+\\!{\,{\rm i}\,}(-{\,{\rm
i}\,}\delta_{{\cal O}}\varsigma^{\hat{{\cal O}}}(x)).\ $ (99)

From this, the field strengths end up transforming homogeneously (as a
curvature), as they would with global (rigid) transformations:

$\displaystyle(\lambda^{\hat{\cal O}}{\bf F}_{\ {\cal O}{\cal
O}^{\prime}}^{\hat{\cal O}})\mapsto(\lambda^{\hat{\cal O}}{\bf F}_{\ {\cal
O}{\cal O}^{\prime}}^{\hat{\cal O}})^{\prime}=(\lambda^{\hat{\cal O}}{\bf
F}_{\ {\cal O}{\cal O}^{\prime}}^{\hat{\cal O}})+\sum_{\tilde{{\cal
O}}}{\,{\rm i}\,}{\varsigma}^{\tilde{{\cal O}}}(x)[\tilde{{\cal
O}},\lambda^{\hat{\cal O}}{\bf F}_{\ {\cal O}{\cal O}^{\prime}}^{\hat{\cal
O}}]_{1}.\ \ $ (100)

Using the assignment of fields in previous sections, and dropping
tensor–products to alleviate notation:

$\displaystyle{D}_{{\cal O}}=\\!{\,{\rm i}\,}(-{\,{\rm i}\,}\delta_{{\cal
O}})+{\,{\rm i}\,}\ell(\hat{\Phi}_{{\cal O}}^{\,\,\tau}{T_{\tau}}+\Phi_{{\cal
O}}^{\,\,\tau}\hat{T}_{\tau})+{\,{\rm i}\,}hB_{{\cal O}}T_{[o]}+{\,{\rm
i}\,}\hat{h}\hat{B}_{{\cal O}}\hat{T}_{[o]}\hskip 42.67912pt$
$\displaystyle\hskip 42.67912pt+{\,{\rm
i}\,}r^{\prime}B^{\,\,[\rho\sigma]}_{{\cal O}}M_{\rho\sigma}+{\,{\rm
i}\,}rB^{\,\,(\rho\sigma)}_{{\cal O}}T_{\rho\sigma},$ (101)
$\displaystyle\,\\!\\![{D}_{{\cal O}},{D}_{{\cal O}^{\prime}}]-{\,{\rm
i}\,}{D}_{[{\cal O},{\cal O}^{\prime}]}\\!=\\!{\,{\rm i}\,}h\,{F}_{{\cal
O}{\cal O}^{\prime}}\,T_{[o]}\\!+\\!{\,{\rm i}\,}\hat{h}\,\hat{{F}}_{{\cal
O}{\cal O}^{\prime}}\,\hat{T}_{[o]}+{\,{\rm i}\,}\ell\,\hat{{F}}_{\ {\cal
O}{\cal O}^{\prime}}^{\rho}\,T_{\rho}+{\,{\rm i}\,}\ell\,{F}_{\ {\cal O}{\cal
O}^{\prime}}^{\rho}\,\hat{T}_{\rho}\hskip 22.76228pt$ $\displaystyle\hskip
42.67912pt+{\,{\rm i}\,}r^{\prime}\,{G}_{\ \ \ {\cal O}{\cal
O}^{\prime}}^{[\tau\kappa]}M_{\tau\kappa}+{\,{\rm i}\,}r{H}_{\ \ \ {\cal
O}{\cal O}^{\prime}}^{(\tau\kappa)}T_{\tau\kappa}.\\!\\!$ (102)

Additional contributions appear from the previously obtained for generic
tilded field strengths appear – compare with (23)– due to the
non–commutativity of differential operators in (93). These new terms generate
quadratic mass terms.

The action in (96) on the fields and field strengths are given by the
expressions in (33–52) by replacing the index $\kappa$ by ${\cal O}$; and
similar expressions changing fields by field strengths. The first order
transformations of the fields ${\cal A}_{{\cal O}}^{\ \hat{\cal O}}$ in (99)
are obtained from (53–58) by replacing the index $\kappa$ by ${\cal O}$, and
changing $\partial_{\kappa}$ by $\delta_{{\cal O}}$. The first order
transformations of the fields strengths ${\bf F}_{\ {\cal O}{\cal
O}^{\prime}}^{\hat{\cal O}}$ in (100) are obtained similarly by changing
fields by field strengths and dopping the inhomogeneous terms (those with
$\delta_{{\cal O}}$).

Additionally, the wider spectrum of covariant derivatives allows for a
different reading of the actions of generators on fields: The transformations
(33-34) and (38) when the label $b$ is a scalar operator ${\cal O}$ represent
rather a mixing between the scalar fields $hB_{\cal O}$, $h\hat{B}_{\cal O}$
and the vector–like fields $\ell\Phi_{\cal O}^{\tau}$, $\ell\bar{\Phi}_{\cal
O}^{\tau}$, which embody the aimed transformations in (1-3) for the case with
just $U(1)$–symmetry.

As we see, derivative-laden terms appear in each $\delta_{{\cal O}}$ in the
field strength. This is not unexpected from the classical point of view, for
instance, rotational contributions to the kinetic term are expected. The field
strengths ${\bf F}_{\ {\cal O}{\cal O}^{\prime}}^{\hat{\cal O}}$ have upper
index $\hat{\cal O}$ which transforms as in (99) and whose basis is the
interaction basis, hence the coupling constants. The lower indices ${\cal
O}{\cal O}^{\prime}$ are those associated to the covariant derivatives and are
not in the interaction basis. To construct an invariant we contract the upper
indices $\hat{\cal O},\hat{\cal O}^{\prime}$ using besides the Minkowski
metric (for lowering or rising indices) also the coefficients of $M^{-1}$ in
(63) depending on the index type completing a matrix ${\bf M}^{-1}_{\hat{\cal
O}\hat{\cal O}^{\prime}}$. For contracting lower indices ${\cal O},{\cal
O}^{\prime\prime}$, besides the Minkowski metric for lowering or rising
Lorentz indices, also use coefficients of $M$ in (2) completing a matrix ${\bf
M}^{{\cal O}{\cal O}^{\prime\prime}}$. The dual or adjoint basis $v^{\cal
O}={\bf M}^{{\cal O}{\cal O}^{\prime\prime}}{\cal O}^{\prime\prime}$
transforms $[\hat{\cal O},v^{\cal O}]=-{\,{\rm i}\,}C_{\ \hat{\cal
O}\tilde{\cal O}}^{{\cal O}}v^{\tilde{\cal O}}$ with the adjoint
representation as does the field strength. From the invariance of $v^{\cal
O}{\bf M}^{-1}_{{\cal O}{\cal O}^{\prime}}v^{{\cal O}^{\prime}}$ it follows
$C^{\tilde{\cal O}}_{\ \hat{\cal O}{\cal O}}{\bf M}^{-1}_{\tilde{\cal O}{\cal
O}^{\prime}}=-{\bf M}^{-1}_{{\cal O}\tilde{\cal O}}C^{\tilde{\cal O}}_{\
\hat{\cal O}{\cal O}^{\prime}}$, making the array $C_{{\cal O}\hat{\cal
O}{\cal O}^{\prime}}$ totally antisymmetric. Hence, using the dual counterpart
formalism to build a globally invariant Lagrangian, we obtain:

$\displaystyle\\!\\!\\!\\!\\!\\!4{L}_{o}\\!=\\!{\,{\rm
i}\,}\\!\lambda^{\hat{\cal O}}{\bf F}_{\ {\cal O}{\cal O}^{\prime}}^{\hat{\cal
O}}{\bf M}^{-1}_{\hat{\cal O}\hat{\cal O}^{\prime}}{\bf M}^{{\cal O}{\cal
O}^{\prime\prime}}{\bf M}^{{\cal O}^{\prime}{\cal
O}^{\prime\prime\prime}}\\!{\,{\rm i}\,}\\!\lambda^{\hat{\cal O}^{\prime}}{\bf
F}_{\ {\cal O}^{\prime\prime}{\cal O}^{\prime\prime\prime}}^{\hat{\cal
O}^{\prime}}\\!=\\!-(\lambda^{\hat{\cal O}})^{2}{\bf F}_{\ {\cal O}{\cal
O}^{\prime}}^{\hat{\cal O}}{\bf F}^{\ {\cal O}{\cal O}^{\prime}}_{\hat{\cal
O}}=\\!\\!\\!\\!$ $\displaystyle\\!\\!\\!\\!\\!\\!{\,{\rm
i}\,}[\ell{{\bar{F}}}_{\ {\cal O}{\cal
O}^{\prime}}^{\kappa}g_{\kappa\kappa^{\prime}}\ \ell{{F}}_{\ {\cal O}{\cal
O}^{\prime}}^{\kappa}g_{\kappa\kappa^{\prime}}\ r^{\prime}{{G}}_{\ \ \ {\cal
O}{\cal
O}^{\prime}}^{[\rho\tau]}\\!g_{\rho\rho^{\prime}}g_{\tau\tau^{\prime}}\
r{{H}}_{\ \ \ {\cal O}{\cal O}^{\prime}}^{(\rho\tau)}\ h{{F}}_{{\cal O}{\cal
O}^{\prime}}\\!g_{\rho\rho^{\prime}}g_{\tau\tau^{\prime}}\
\hat{h}{{\hat{F}}}_{{\cal O}{\cal O}^{\prime}}]\cdot\hskip 28.45274pt\ $
$\displaystyle M^{-1}\cdot{\,{\rm i}\,}[\ell{{F}}_{\ {\cal
O}^{\prime\prime}{\cal O}^{\prime\prime\prime}}^{\kappa^{\prime}}\
\ell{{\bar{F}}}_{\ {\cal O}^{\prime\prime}{\cal
O}^{\prime\prime\prime}}^{\kappa^{\prime}}\ r^{\prime}{{G}}_{\ \ \ {\cal
O}^{\prime\prime}{\cal
O}^{\prime\prime\prime}}^{[\rho^{\prime}\tau^{\prime}]}\ r{{H}}_{\ \ \ {\cal
O}^{\prime\prime}{\cal
O}^{\prime\prime\prime}}^{(\rho^{\prime}\tau^{\prime})}\ h{{F}}_{{\cal
O}^{\prime\prime}{\cal O}^{\prime\prime\prime}}\ \hat{h}{{\hat{F}}}_{{\cal
O}^{\prime\prime}{\cal O}^{\prime\prime\prime}}]^{tr}\\!\\!\\!\\!$
$\displaystyle{\bf M}^{{\cal O}{\cal O}^{\prime\prime}}\\!{\bf M}^{{\cal
O}^{\prime}{\cal O}^{\prime\prime\prime}},\\!\\!$ $\displaystyle
4L_{o}=\left(-\frac{\epsilon\ell^{2}}{2}\\{{{\bar{F}}}^{\ {\cal O}{\cal
O}^{\prime}}_{\kappa},{{F}}_{\ {\cal O}{\cal
O}^{\prime}}^{\kappa}\\}-{r^{\prime 2}}{{G}}^{\ \ \ {\cal O}{\cal
O}^{\prime}}_{[\rho\tau]}{{G}}_{\ \ \ {\cal O}{\cal
O}^{\prime}}^{[\rho\tau]}-{r^{2}}{{H}}^{{\cal O}{\cal O}^{\prime}}_{\ \
(\rho\tau)}{{H}}_{{\cal O}{\cal O}^{\prime}}^{\ \ (\rho\tau)}\right.$
$\displaystyle\hskip 56.9055pt\left.-\frac{5h^{2}}{24}{{F}}^{{\cal O}{\cal
O}^{\prime}}{{F}}_{{\cal O}{\cal O}^{\prime}}-{5\hat{h}^{2}}{{\hat{F}}}^{{\cal
O}{\cal O}^{\prime}}{\hat{F}}_{{\cal O}{\cal O}^{\prime}}\right).$ (103)

The local invariance of the action with Lagrangian ${L}_{o}$ follows from the
fact that field strengths transform homogeneously, and the quadratic casimir
structure of the action does the rest. So, we have for instance
$[\ell\,T_{\mu},\int\cdots\int{L}_{o}dx^{n}]=0$.

##  6 Equations of Motions & Gauge Invariance

First we should notice that our group does not have generators for actual
plain translations in the adopted extended space–time, nor any device has been
set up yet to gain them after say a contraction mechanism or some analogous
procedure. So, we are in a stage where plain space–time translations are not
allowed.

Let $F(x)$, with $(x)$ standing for the coordinates involved in the adopted
differential representation $\\{-{\,{\rm i}\,}\delta_{{\cal O}}\,|\,{{\cal
O}}$ operator of the extended Lorentz symmetry$\\}$, be a differentiable
function of its arguments, and that vanish or is exponentially decaying on
each of the limits of the integration of the action
$\int\cdots\int{L}_{o}dx^{n}$. Let $\tilde{\varsigma}(x)$ be an arbitrary
function of the geometric–frame coordinates. For the obtained differential
representations we have:

$\displaystyle\int\cdots\int F(x)\,(-{\,{\rm i}\,}\delta_{{\cal
O}"}\tilde{\varsigma}(x))\,d\chi^{n}=-\int\cdots\int(-{\,{\rm
i}\,}\delta_{{\cal O}"}F(x))\tilde{\varsigma}(x)\,d\chi^{n}$
$\displaystyle+({\rm vanishing\ boundary\ contributions}).$ (104)

This property holds for the differential representations in the previous
section independently of the typology of the $-{\,{\rm i}\,}\delta_{{\cal
O}"}$, be it hermitic, anti-hermitic, or member of an adjoint pair.

Now, ${{L}_{o}}(\lambda^{\hat{\cal O}}{{\cal A}}_{{\cal O}}^{\ \hat{\cal
O}}(x),-{\,{\rm i}\,}\delta_{\tilde{\cal O}}\lambda^{\hat{\cal O}}{{\cal
A}}_{{\cal O}}^{\ \hat{\cal O}}(x))$ is a differentiable function of its
arguments, and it vanishes for vanishing arguments. And we assume that both
the fields $\lambda^{\hat{\cal O}}{{\cal A}}_{{\cal O}}^{\ \hat{\cal O}}(x)$
(accompanied with its coupling constant), and $-{\,{\rm
i}\,}\delta_{\tilde{\cal O}}\lambda^{\hat{\cal O}}{{\cal A}}_{{\cal O}}^{\
\hat{\cal O}}(x)$ vanish or are exponentially decaying on each of the
integration limits of the action $A_{o}=\int\cdots\int{L}_{o}d\chi^{n}$. We
use (104) and the variational analysis notation

$\frac{\delta A_{o}}{\delta(\lambda^{\hat{\cal O}}{{\cal A}}_{{\cal O}}^{\
\hat{\cal O}}(x))}=\frac{\partial{{L}_{o}}}{\partial(\lambda^{\hat{\cal
O}}{{\cal A}}_{{\cal O}}^{\ \hat{\cal O}}(x))}-\left(-{\,{\rm
i}\,}\delta_{\tilde{\cal O}}\,\frac{\partial{{L}_{o}}}{\partial(-{\,{\rm
i}\,}\delta_{\tilde{\cal O}}\lambda^{\hat{\cal O}}{{\cal A}}_{{\cal O}}^{\
\hat{\cal O}}(x))}\right).$ (105)

We have ${L}_{o}=\frac{1}{4}({\,{\rm i}\,}\lambda^{\hat{\cal O}}{\bf F}_{\
{\cal O}{\cal O}^{\prime}}^{\hat{\cal O}})({\,{\rm i}\,}\lambda_{\hat{\cal
O}}{\bf F}^{\ {\cal O}{\cal O}^{\prime}}_{\hat{\cal O}})$. The condition of a
vanishing variation of $A_{o}$ under arbitrary field variations (consistent
with their conditions at the integration limits) becomes (in the first line we
use (105)):

$\displaystyle\delta A_{o}=\int\cdots\int\left(\delta(\lambda^{\hat{\cal
O}}{{\cal A}}_{{\cal O}}^{\ \hat{\cal O}}(x))\frac{\delta
A_{o}}{\delta(\lambda^{\hat{\cal O}}{{\cal A}}_{{\cal O}}^{\ \hat{\cal
O}}(x))}\right)d\chi^{n}+({\rm v.\ b.\ c.})\\!\\!\\!$
$\displaystyle\\!=\\!\\!\int\cdots\int\delta(\lambda^{\hat{\cal O}}{{\cal
A}}_{{\cal O}}^{\ \hat{\cal O}}(x))\left(\frac{\partial(\lambda^{\hat{\cal
O}}{\bf F}_{\ {\cal O}{\cal O}^{\prime}}^{\hat{\cal
O}})}{\partial(\lambda^{\hat{\cal O}}{{\cal A}}_{{\cal O}}^{\ \hat{\cal
O}}(x))}\frac{\partial{{L}_{o}}\ \ \ }{\partial(\lambda^{\hat{\cal O}}{\bf
F}_{\ {\cal O}{\cal O}^{\prime}}^{\hat{\cal O}})}\right.$
$\displaystyle\left.-(-{\,{\rm i}\,}\delta_{\tilde{\cal
O}})\left(\,\frac{\partial(\lambda^{\hat{\cal O}}{\bf F}_{\ {\cal O}{\cal
O}^{\prime}}^{\hat{\cal O}})\ \ }{\partial(-{\,{\rm i}\,}\delta_{\tilde{\cal
O}}\lambda^{\hat{\cal O}}{{\cal A}}_{{\cal O}}^{\ \hat{\cal
O}}(x))}\frac{\partial{{L}_{o}}\ \ \ }{\partial(\lambda^{\hat{\cal O}}{\bf
F}_{\ {\cal O}{\cal O}^{\prime}}^{\hat{\cal
O}})}\right)\right)\\!d\chi^{n}+({\rm v.\ b.\ c.})=0,\\!\\!\\!$

where v. b. c. stands for vanishing boundary contributions. The equation of
motion guarantees the invariance of the action for generic field variations:

$\,[D_{{\cal O}^{\prime}}]_{\hat{\cal O}^{\prime}}^{\ \ \hat{\cal
O}}\left({\lambda_{\hat{\cal O}}}{\bf F}_{\hat{\cal O}}^{\ {\cal O}{\cal
O}^{\prime}}\right)-{\,{\rm i}\,}({\,{\rm i}\,}C_{\ \bar{\cal O}{\cal
O}^{\prime}}^{{\cal O}})\left({\lambda_{\hat{\cal O}^{\prime}}}{\bf
F}_{\hat{\cal O}^{\prime}}^{\ \bar{\cal O}{\cal O}^{\prime}}\right)=0.$ (106)

In terms of the field strength ${\bf F}_{\ {\cal O}{\cal
O}^{\prime}}^{\hat{\cal O}}$ the equation of motion becomes

$\displaystyle\,[D^{{\cal O}^{\prime}}]^{\hat{\cal O}^{\prime}}_{\ \ \hat{\cal
O}}\left({\lambda^{\hat{\cal O}}}{\bf F}^{\hat{\cal O}}_{\ {\cal O}{\cal
O}^{\prime}}\right)-{\,{\rm i}\,}({\,{\rm i}\,}C^{\ \bar{\cal O}{\cal
O}^{\prime}}_{{\cal O}})\left({\lambda^{\hat{\cal O}^{\prime}}}{\bf
F}^{\hat{\cal O}^{\prime}}_{\ \bar{\cal O}{\cal O}^{\prime}}\right)=0,\hskip
56.9055pt$ (107) $\displaystyle{\rm where}\ [D^{{\cal O}^{\prime}}]^{\hat{\cal
O}^{\prime}}_{\ \ \hat{\cal O}}=\delta^{{\cal O}^{\prime}}[Id]^{\hat{\cal
O}^{\prime}}_{\ \ \hat{\cal O}}+{\,{\rm i}\,}(\lambda_{\tilde{\cal O}}{\cal
A}^{{\cal O}^{\prime}}_{\ \tilde{\cal O}})[R^{\tilde{\cal O}}]^{\hat{\cal
O}^{\prime}}_{\ \ \hat{\cal O}},\ [R^{\tilde{\cal O}^{\prime}}]^{\hat{\cal
O}^{\prime}}_{\ \ \hat{\cal O}}=-{\,{\rm i}\,}C^{\ \tilde{\cal O}\hat{\cal
O}^{\prime}}_{\hat{\cal O}},\ $ (108)

lowering and rising of indices are done using ${\bf M}^{-1}_{\hat{\cal
O}\hat{\cal O}^{\prime}}$ and ${\bf M}^{{\cal O}{\cal O}^{\prime\prime}}$
respectively. Notice, $C^{\ \tilde{\cal O}\hat{\cal O}^{\prime}}_{\hat{\cal
O}}$ is the structure constant of the dual basis algebra, $v^{\cal O}={\bf
M}^{{\cal O}{\cal O}^{\prime\prime}}{\cal O}^{\prime\prime}$.

From the equation of motion (106) it is easy to propose a conserved current,
where the last equality in (109) uses the equation of motion:

$\displaystyle\\!\\!{\cal J}_{\hat{\cal O}}^{\ \bar{\cal O}}=\delta_{{\cal
O}^{\prime}}(\lambda_{\hat{\cal O}}{\bf F}_{\hat{\cal O}}^{\ \bar{\cal O}{\cal
O}^{\prime}})-\frac{{\,{\rm i}\,}}{2}({\,{\rm i}\,}C_{\ {\cal O}{\cal
O}^{\prime}}^{\bar{\cal O}})(\lambda_{\hat{\cal O}}{\bf F}_{\hat{\cal O}}^{\
{\cal O}{\cal O}^{\prime}})$ $\displaystyle\ \ =-{\,{\rm
i}\,}(\lambda^{\hat{\cal O}^{\prime}}{{\cal A}}_{{\cal O}^{\prime}}^{\
\hat{\cal O}^{\prime}})(-{\,{\rm i}\,}C_{\ \hat{\cal O}^{\prime}\hat{\cal
O}}^{\tilde{\cal O}})(\lambda_{\tilde{\cal O}}{\bf F}_{\tilde{\cal O}}^{\
\bar{\cal O}{\cal O}^{\prime}})+\frac{{\,{\rm i}\,}}{2}({\,{\rm i}\,}C_{\
{\cal O}{\cal O}^{\prime}}^{\bar{\cal O}})(\lambda_{\hat{\cal O}}{\bf
F}_{\hat{\cal O}}^{\ {\cal O}{\cal O}^{\prime}}),\ \ \ \ $ (109)
$\displaystyle\delta_{\bar{\cal O}}{\cal J}_{\hat{\cal O}}^{\ \bar{\cal
O}}=0.$ (110)

We can also define a covariant conserved current, where the last equality in
(111) uses the equation of motion:

$\displaystyle\\!\\!{J}_{\hat{\cal O}}^{\ \bar{\cal O}}=[D_{{\cal
O}^{\prime}}]_{\hat{\cal O}}^{\ \tilde{\cal O}}(\lambda_{\tilde{\cal O}}{\bf
F}_{\tilde{\cal O}}^{\ \bar{\cal O}{\cal O}^{\prime}})\\!-\\!\frac{{\,{\rm
i}\,}}{2}({\,{\rm i}\,}C_{\ {\cal O}{\cal O}^{\prime}}^{\bar{\cal
O}})(\lambda_{\hat{\cal O}}{\bf F}_{\hat{\cal O}}^{\ {\cal O}{\cal
O}^{\prime}})=\frac{{\,{\rm i}\,}}{2}({\,{\rm i}\,}C_{\ {\cal O}{\cal
O}^{\prime}}^{\bar{\cal O}})\lambda_{\hat{\cal O}}{\bf F}_{\hat{\cal O}}^{\
{\cal O}{\cal O}^{\prime}}\\!\\!,$ (111) $\displaystyle\,[D_{\bar{\cal
O}}]_{\hat{\cal O}}^{\ \tilde{\cal O}}{J}_{\tilde{\cal O}}^{\ \bar{\cal
O}}=0.\hskip 227.62204pt$ (112)

Now, the dynamic of our model thus far is governed by the underlying group and
algebra. We know which is the transformation of fields and their corresponding
field strengths under local group actions. We consider the action of
differential representations on the extended space–time, constraining them to
produce a smooth bijection, so that the change of variables theorem apply:

$\displaystyle\chi^{a}$ $\displaystyle\mapsto$ $\displaystyle\chi^{a}-{\,{\rm
i}\,}\vartheta\varsigma^{\tilde{\cal O}}(x)(-{\,{\rm i}\,}\delta_{\tilde{\cal
O}}\chi^{a}),$ (113) $\displaystyle d\chi^{a}$ $\displaystyle\mapsto$
$\displaystyle d\chi^{a}+\partial_{b}(-{\,{\rm
i}\,}\vartheta\varsigma^{\tilde{\cal O}}(x)(-{\,{\rm i}\,}\delta_{\tilde{\cal
O}}\chi^{a}))d\chi^{b},$ $\displaystyle d\chi^{n}$ $\displaystyle\mapsto$
$\displaystyle d\chi^{n}+\partial_{a}(-{\,{\rm
i}\,}\vartheta\varsigma^{\tilde{\cal O}}(x)(-{\,{\rm i}\,}\delta_{\tilde{\cal
O}}\chi^{a}))d\chi^{n}.$ (114)

Notice that only diagonal contributions contribute in first order to the full
volume measure $d\chi^{n}$. We can turn on and off the action of the
differential representation taking $\vartheta=1$ or $\vartheta=0$. We examine
dynamic conditions that assure the first order invariance of the action under
local gauge transformations (for $\vartheta=0$), or a combination of both the
differential representation and local gauge transformations (for
$\vartheta=1$). In (115), the field $\lambda^{\hat{\cal O}}{{\cal A}}_{{\cal
O}}^{\ \hat{\cal O}}(x)$ transforms as $\lambda^{\hat{\cal O}}{{\cal
A}}_{{\cal O}}^{\ \hat{\cal O}}(x)\mapsto\lambda^{\hat{\cal O}}{{\cal
A}}_{{\cal O}}^{\ \hat{\cal O}}(x)+\delta(\lambda^{\hat{\cal O}}{{\cal
A}}_{{\cal O}}^{\ \hat{\cal O}}(x))$ with

$\displaystyle\delta(\lambda^{\hat{\cal O}}{{\cal A}}_{{\cal O}}^{\ \hat{\cal
O}}(x))=\sum_{\tilde{\cal O}}{\,{\rm i}\,}\vartheta{{\varsigma}^{\tilde{\cal
O}}(x)}(\\!-(-{\,{\rm i}\,}\delta_{\tilde{\cal O}}(\lambda^{\hat{\cal
O}}{{\cal A}}_{{\cal O}}^{\ \hat{\cal O}}(x))))\hskip 56.9055pt$
$\displaystyle\hskip 56.9055pt+\sum_{\tilde{\cal O}}{\,{\rm
i}\,}{{\varsigma}^{\tilde{\cal O}}(x)}[\tilde{\cal O},\lambda^{\hat{\cal
O}}{{\cal A}}_{{\cal O}}^{\ \hat{\cal O}}(x)]_{1}+{\,{\rm i}\,}(-{\,{\rm
i}\,}\delta_{{\cal O}}\varsigma^{\hat{{\cal O}}}(x)).$ (115)

Using (104), the condition of a vanishing variation of $A_{o}$ under local
gauge transformations (115) and (114) becomes:

$\displaystyle\int\cdots\int\left(\delta(\lambda^{\hat{\cal O}}{{\cal
A}}_{{\cal O}}^{\ \hat{\cal O}}(x))\frac{\delta
A_{o}}{\delta(\lambda^{\hat{\cal O}}{{\cal A}}_{{\cal O}}^{\ \hat{\cal
O}}(x))}+L_{o}\partial_{a}(-{\,{\rm i}\,}\vartheta\varsigma^{\tilde{\cal
O}}(x)(-{\,{\rm i}\,}\delta_{\tilde{\cal O}}x^{a}))\right)d\chi^{n}=\\!\\!\\!$
$\displaystyle\int\cdots\int\left(\delta(\lambda^{\hat{\cal O}}{{\cal
A}}_{{\cal O}}^{\ \hat{\cal O}}(x))\frac{\delta
A_{o}}{\delta(\lambda^{\hat{\cal O}}{{\cal A}}_{{\cal O}}^{\ \hat{\cal
O}}(x))}-\partial_{a}(L_{o})(-{\,{\rm i}\,}\vartheta\varsigma^{\tilde{\cal
O}}(x)(-{\,{\rm i}\,}\delta_{\tilde{\cal O}}x^{a}))\right)d\chi^{n}\\!\\!\\!$
$\displaystyle+({\rm vanishing\ boundary\ contributions})=$
$\displaystyle\int\cdots\int\left(\delta(\lambda^{\hat{\cal O}}{{\cal
A}}_{{\cal O}}^{\ \hat{\cal O}}(x))\frac{\delta
A_{o}}{\delta(\lambda^{\hat{\cal O}}{{\cal A}}_{{\cal O}}^{\ \hat{\cal
O}}(x))}-(-{\,{\rm i}\,}\vartheta\varsigma^{\tilde{\cal O}}(x))(-{\,{\rm
i}\,}\delta_{\tilde{\cal O}}(L_{o}))\right)d\chi^{n}\ \ $ $\displaystyle+({\rm
vanishing\ boundary\ contributions})=0.\ \ $ (116)

Now, we have ${L}_{o}=({\,{\rm i}\,}\\!\lambda^{\hat{\cal O}})^{2}{\bf F}_{\
{\cal O}{\cal O}^{\prime}}^{\hat{\cal O}}{\bf F}^{\ {\cal O}{\cal
O}^{\prime}}_{\hat{\cal O}}$. The variation of field strengths
$(\lambda^{\hat{\cal O}}{\bf F}_{\ {\cal O}{\cal O}^{\prime}}^{\hat{\cal O}})$
$\mapsto(\lambda^{\hat{\cal O}}{\bf F}_{\ {\cal O}{\cal
O}^{\prime}}^{\hat{\cal O}})+\delta(\lambda^{\hat{\cal O}}{\bf F}_{\ {\cal
O}{\cal O}^{\prime}}^{\hat{\cal O}})$ is:

$\displaystyle\\!\\!\\!\delta(\lambda^{\hat{\cal O}}{\bf F}_{\ {\cal O}{\cal
O}^{\prime}}^{\hat{\cal O}})\\!\\!$
$\displaystyle\\!\\!\\!=\\!\sum_{\tilde{\cal O}}\left(\\!-{\,{\rm
i}\,}\vartheta{{\varsigma}^{\tilde{\cal O}}\\!(x)}(-{\,{\rm
i}\,}\delta_{\tilde{\cal O}}(\lambda^{\hat{\cal O}}{\bf F}_{\ {\cal O}{\cal
O}^{\prime}}^{\hat{\cal O}})\\!\\!\right)\\!$ (117)
$\displaystyle+\\!\sum_{\tilde{\cal O}}\\!{\,{\rm
i}\,}{{\varsigma}^{\tilde{\cal O}}\\!(x)}[\tilde{\cal O},{\lambda^{\hat{\cal
O}}}{\bf F}_{\ {\cal O}{\cal O}^{\prime}}^{\hat{\cal O}}]_{1}.\ \ \ $

Hence, the condition for vanishing variation of the action $A_{o}$ becomes

$\displaystyle\int\\!\cdots\\!\int\\!\left(\delta(\lambda^{\hat{\cal O}}{\bf
F}_{\ {\cal O}{\cal O}^{\prime}}^{\hat{\cal O}})\frac{\delta
A_{o}}{\delta(\lambda^{\hat{\cal O}}{\bf F}_{\ {\cal O}{\cal
O}^{\prime}}^{\hat{\cal O}})}-(-{\,{\rm i}\,}\vartheta\varsigma^{\tilde{\cal
O}}(x))(-{\,{\rm i}\,}\delta_{\tilde{\cal O}}(L_{o})\right)d\chi^{n}$
$\displaystyle=\int\\!\cdots\\!\int\\!\left(\left(\\!-{\,{\rm
i}\,}\vartheta{{\varsigma}^{\tilde{\cal O}}(x)}\\!(-{\,{\rm
i}\,}\delta_{\tilde{\cal O}}(\lambda^{\hat{\cal O}}{\bf F}_{\ {\cal O}{\cal
O}^{\prime}}^{\hat{\cal O}}))\\!+\\!\\!{\,{\rm i}\,}{{\varsigma}^{\tilde{\cal
O}}\\!(x)}(-{{\,{\rm i}\,}}C^{\ \hat{\cal O}}_{{\tilde{\cal O}},{{\cal
O}^{\prime\prime}}}\lambda^{{\cal O}^{\prime\prime}}{\bf F}_{\ {\cal O}{\cal
O}^{\prime}}^{{\cal O}^{\prime\prime}})\right)\right.\cdot$
$\displaystyle\hskip 28.45274pt\cdot\left(\\!\frac{2{\,{\rm
i}\,}}{4}\lambda_{\hat{\cal O}}({\,{\rm i}\,}{\bf F}^{\ {\cal O}{\cal
O}^{\prime}}_{\hat{\cal O}})\\!\right)\left.-(\\!-{\,{\rm
i}\,}\vartheta\varsigma^{\tilde{\cal O}}(x))(-{\,{\rm i}\,}\delta_{\tilde{\cal
O}}(L_{o})\right)d\chi^{n}=0.$ (118)

The term associated to the first term in the first factor in the right-hand
side of (118) ends up canceling with the term coming from the measure
variation (114), both with the factor $\vartheta$, essentially the change of
variables theorem. The term with structure constant cancels due to the
invariance of $L_{o}$ under global transformations. So, for arbitrary
$\varsigma^{{\cal O}}(x)$ –as long as the coordinate transformations remain a
biyection– we get invariance of the action $A_{o}$. Notice that we do not need
the equations of motion for this invariance. Hence, under the given
assumptions, both local gauge symmetry and transformations associated to the
action of covariant derivatives are automatically satisfied. The picture that
emerges is that, at least mathematically, the model genesis stems from a Lie
group manifold with its invariant Haar metric, which after a series of
transmutations associated to contraction procedures, topological
compactifications or plain symmetry breakings gets gradually closer to the
standard model scenery.

##  7 Alternative Considerations

An alternative to the obtained non–contracted Lorentz extension model could be
found if we entertain the notion where we want to direct the role of
generators $T_{\mu},\bar{T}_{\nu}$ to be translations after a further process
of contraction (after having performed the first round of contraction getting
rid of the $x^{4}$). If we wanted to profile such a model as a forerunner of a
gravity theory, the coupling constant $\ell$ could take the form
$\lambda/\Lambda$, where $\Lambda$ is a dimension -1 parameter (a length)
associated to a possible contraction, and $\lambda$ will be a dimensionless
coupling constant. In this case, we could view the fields $\Phi_{a}^{\
\mu},\bar{\Phi}_{a}^{\ \mu}$ as dimensionless Vielbein fields, and the
combination ${\bf g}_{ab}=\Phi_{a}^{\,\mu}\bar{\Phi}_{b\mu}$ will be a
prospective complex metric (the symmetric part being real, the antisymmetric
imaginary). But such a metric is only invariant in the contracted limit, which
needs to be performed. Hence, a term analogous to $\sqrt{-det({\bf g})}$ would
be obtained from $(1/4!)\epsilon_{\mu\nu\rho\sigma}\Phi_{a}^{\
\mu}\hat{\Phi}_{b}^{\ \nu}\Phi_{c}^{\ \rho}\hat{\Phi}_{s}^{\
\sigma}\epsilon^{abcs}$. Alternatively, in order to obtain a term like this
from an actual global invariant prior to contraction, we appeal to the
non–homogeneous hollow casimir (21). Using the similar transformations
properties for field and field strenths, we could propose as a forerunner of
the dimensionless $\sqrt{-det({\bf g})}$ the lagrangian density factor of the
form:

$\displaystyle\epsilon_{\mu\nu\rho\sigma}\left({\frac{\lambda^{4}\epsilon^{2}}{4}}\Phi_{a}^{\
\mu}\hat{\Phi}_{b}^{\ \nu}\Phi_{c}^{\ \rho}\hat{\Phi}_{s}^{\
\sigma}+\Lambda^{4}{r^{\prime 2}}{{G}}^{[\mu\nu]}_{\ \ \
ab}{{G}}^{[\rho\sigma]}_{\ \ \ cs}\right)(1/4!)\epsilon^{abcs}.$ (119)

It is not clear however, that this term can compensate the variation of the
measure $dx^{n}$ under some of the transformations. We need to examine each
symmetry transformation and the corresponding equation of motion separately.
On the other hand, the degrees of freedom associated with fields $\Phi_{a}^{\
\mu},\hat{\Phi}_{a}^{\ \mu}$ need to be matched with a prospective complex
scalar for the Higgs field and a complex metric. These considerations will be
addressed when considering the Poincaré–like extensions in [7]. There, we
include separately the forerunners of translations prior to
$x^{4}$–contraction. We will have a homogeneous hollow casimir (identity),
which can involve a mixture of a candidate for Higgs with the Vielbein
candidates for gravity.

Consider now a model symmetric under the 10D algebra
$<\\!\\{M_{\mu\nu},T_{\rho}\\!+\\!\hat{T}_{\rho}\\}\\!>$. This will entail the
restriction to the corresponding covariant derivative with: real field
$\tilde{\Phi}_{a}^{\ \rho}$ with coupling constant $\lambda/\Lambda$
associated to $T_{\rho}+\hat{T}_{\rho}$, and real field $B_{a}^{\,[\mu,\nu]}$
with coupling constant $r^{\prime}$ associated to $M_{\mu\nu}$. The quadratic
casimir (18) leads to a globally invariant lagrangian density along the same
lines as done for the full LE. In this case, the generators involved in the
hollow casimir (20) does include all symmetry generators. So, by taking as the
prospective metric ${\bf g}_{ab}=\tilde{\Phi}_{a}^{\,\mu}\tilde{\Phi}_{b\mu}$,
we could propose as a forerunner of the dimensionless $\sqrt{-det({\bf g})}$
the lagrangian density factor of the form:

$\displaystyle\epsilon_{\mu\nu\rho\sigma}\left({{\lambda^{4}\epsilon^{2}}}\tilde{\Phi}_{a}^{\
\mu}\tilde{\Phi}_{b}^{\ \nu}\tilde{\Phi}_{c}^{\ \rho}\tilde{\Phi}_{s}^{\
\sigma}+\Lambda^{4}{r^{\prime 2}}{{G}}^{[\mu\nu]}_{\ \ \
ab}{{G}}^{[\rho\sigma]}_{\ \ \ cs}\right)(1/4!)\epsilon^{abcs}.$ (120)

The globally invariant factors presented used $(1/4!)\epsilon^{abcs}$ to
condensate the covariant derivative indices, but it might be omited, and set
$a=0,\cdots,s=3$ that connect to the labels in the measure $dx^{n}$. There are
reasons of concern in using a non–homogeneous invariant identity.

Now, the contraction procedure used to arrive to the Poincaré algebra can also
be considered for the variables involved in the space–time complexification.
We could set ${\bf T}_{\mu\nu}=(1/\Lambda^{2}){T}_{\mu\nu},{\bf
T}_{\rho}=(1/\Lambda)T_{\rho},\hat{\bf T}_{\rho}=(1/\Lambda)\hat{T}_{\rho}$,
which in the end corresponds to the replacement
$r=\tilde{r}/\Lambda^{2},\ell=\lambda/\Lambda$. In fact, the contraction
formalism, although a well posed mathematical procedure for undergoing from a
Lie algebra to another, does not necessarily account for the actual physical
underpinnings for such procedure. The nature of the gravitational coupling
naively recalls the Fermi constant, which just account for the effective
remnant of a massive gauge field. We do not know if this is the case here, but
playing the game of introducing dimensionful couplings (and dimensionless
fields) let us foresee possible sceneries for the path to effective low energy
symmetry.

The building of a factor embodying an invariant measure suggest a quartic
invariant (extending the Pauli-Lubański invariant) in the fields. Some
extension of the quadratic invariant in (91) might lead the ultimate invariant
measure.

##  8 Relations, Outlook, Conclusions

We have launched a program to profit from certain observations on structures
of the SM of particle physics that aimed to obtain symmetries that render
naturalness to them. To simplify matters, we departed from a model with just
an internal $U(1)$ symmetry, and proposed a transformation combining Higgs and
gauge fields inspired in their similar ultraviolet behavior, and their
ultimate recombination during EW symmetry breaking. In attempting to close an
algebra for such a symmetry, we were led to a Lie algebra involving both
internal $U(1)$ and external symmetries, such as Lorentz transformations. This
extension renders by itself novel extensions of external symmetry involving a
traceless spin 2 generator, some symmetric vectors, and scalars. For the
obtained extension of the Lorentz algebra we constructed first a locally
invariant model but without affecting the geometric frame (so, not actually
involving external symmetry), where some gauge fields seem to adopt the role
of a Higgs–like scalar field.

We developed representations of the symmetry in terms of differential
operators to gain some insight into its geometrical counterpart. It also
prepares the path for obtaining the corresponding Poincaré extensions in [7].
We incorporate a differential representation into a model whose geometric
frame is coordinatized by the variables involved in the differential
representation. The differential operators reproduce locally the symmetry, and
the local gauge transformations introduce smooth distortions from such
background. The labels of the covariant derivatives are those of the geometric
manifold. The considered extension do not have operators to be forcefully
promoted to be the ultimate space–time translations unless such a role is
conferred to some of the generators in the extension. Some preliminary
considerations in alternative models gave an exploration of these
possibilities. The application of an extended equivalence principle might
grant the existence of a locally flat or de Sitter space–time measure. Our
equations of motions were deprived from these constructions. We will tackle
more extensively such considerations when dealing with Poincaré–like
extensions, where there are clearly designated translations. Some generators
take over the role of remnant space–time translations at each stage of the
universe evolution.

These constructions reveal that the Lorentz Lie algebra extension is the Lie
algebra of $u(s_{2},t_{2})$ with $s_{2},t_{2}>0$ and $s_{2}+t_{2}=5$. In order
to assign the origin of the Higgs field to certain vector generators,
$t_{2}=1$ is demanded. The obtained extension include as subalgebras some
novel extensions of the Lorentz algebra not considered in the literature
elsewhere. In the differential representations we introduced a prospecting
dimension (here $\chi^{4}$ and perhaps $y^{4}$) since this will enable dealing
to Poincaré extensions in a further exploration in [7]. As pointed out, the
Lorentz Extension (LE) contains readily the seeds for gravity considerations.
We expect, that a linear combination of these translation precursors with the
translations to be introduced in [7] will compose the ultimate translations
and its massless (or close to massless) graviton, while the further
translation–like combination breaks and their graviton–like field acquire
significant mass.

Only a more accurate examination of the diverse possibilities, actual paths
towards the low energy effective model of the Standard Model with gravity, and
the hopefully, stability of involved symmetries under quantum fluctuations
will help to settle which extensions might be more adequate. It is not clear
from the outset, which will be the symmetry breaking path or contraction
procedures leading at low energies to the aimed features of the SM. A
following issue of this program in [7] will obtain the corresponding extension
of the Poincaré symmetry and address the introduction of gravity through the
local invariance under forerunners of space–time translations. How do we
enlarge such simplified models to include isospin ($SU(2)$) and strong
interaction ($SU(3)$) internal symmetries will be tackled in further
contributions, first aiming to include the SM electroweak bosonic sector, and
then tackling larger internal symmetries.

The complexification underlying the so called 10D differential representation
seems to converge in part to several developments leading to similar features.
From the early unification attempts by Einstein and Strauss [9], and E.
Schrödinger [10], to diverse remarkable constructions in the crossroads of
string theory, non–commutative geometry, and gravity theory
[11][12][13][14][15][16][17] [18][19][20][21][22][23][24][25][26][27]. Some
have implemented complexifications to cope with diverse challenges at Planck
scales, compactification, tenets of noncommutative geometry, and
phenomenological results on Higgs mass. In particular, the 15 dimensional
subalgebra $l=<\\{M_{\mu\nu},T_{\rho\sigma}\\}>$ in (10)–(14) could coincide
with the algebra ${\mathfrak{so}}(1,5)$ (or the conformal
${\mathfrak{so}}(2,4)$) used in Yang’s Space–time and its modified version
[28][29], and the one used to construct a fuzzy 4D de Sitter space–time in
[25][27]. The non contracted Lorentz extension provide a novel way to produce
a fuzzy 4D de Sitter space–time. The geometrical frame underlying the
differential representations and model will profit from prominent
constructions dealing with compactification of extra (now hidden) dimensions
[30][31][32]. The present approach seems to extend noncommutative geometry
constructions where the Higgs fields is the gauge field associated to discrete
dimensions (or spectra) and seemingly discrete algebraic structures. Here, the
would–be–Higgs field proceeds from a gauge field of a continuous symmetry. It
might be the case that symmetry breaking, contractions or emerging brane
structures constrain the symmetry associated to such Higgs field to converge
with the tenets of noncommutative geometry models. It is appealing, that
similar or identical substructures have emerged in so diverse settings.

One remark on noncommutative and/or nonassociative coordinates is in place:
The use of noncommutative and even nonassociative coordinates as parameters of
transformations that composes ( via parameter times generator) Lie group
structures can be avoided in some instances. When the noncommutativity and/or
nonassociativity is governed by (discrete) finite gradings compatible with the
Lie algebraic structure, and there exist a finite basis embodying in the
parameters such noncommutative and nonassociative behaviour, then there is a
change of variables (parameters) restoring commutativity and associativity
[33]. Under these hypothesis, this leaves only Grassmann variables
noncommutativity (and the corresponding super Lie algebras structures) as the
only truly nontrivial contributions in the noncommutative and nonassociative
realm. If the referred change of variables can be performed locally in the
space–time, and there is a smooth change of variables for neighboring
space–time points, then the use of noncommutativity and/or nonassociativity
might be avoided if a smooth coordinate transformation could be pasted for the
whole space–time. The existence of transformations maintaining self–duality
needs to be settled separately. Nevertheless, there is to my knowledge, no
general assessment of the possibility of having such variable changes. There
is in Noncommutative Geometry an unexpected richness of structures rendering
naturality to the origin of multiple features of the SM and gravity.

We will pursue an extension of symmetry involving fermionic generators, so
that we can aim to assign fermionic particles to remnants of such generators,
joining early visionary developments by P. Dirac [34]. Some developments in
quantum gravity on entropy have assigned a quantum of area in terms of squared
Planck length to a black hole event horizon [35][36] or a further inner
horizon singularity [37]. It might be that it is the exclusion principle in
place assigning some fundamental fermionic particles to diverse quantum area
locations in the ultimate horizon.

##  9 Acknowledgments

The first stages of this research took place at the Departamento de Física
Teórica y del Cosmos, Universidad de Granada, Spain, and CIMAT, Guanajuato,
México. The Fundación Carolina is acknowledged for the financial support of
the research stay in Spain, and CIMAT is acknowledged for the finantial
support for the research stay in México. The Universidad Nacional de Colombia,
Sede Medellín, is aknowleged for its support of this research.

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