# Quantum doubles of Fock type and bosonization

Dimitry Gurevich
Institute for Information Transmission Problems
Bolshoy Karetny per. 19, Moscow 127051, Russian Federation
Pavel Saponov
National Research University Higher School of Economics,
20 Myasnitskaya Ulitsa, Moscow 101000, Russian Federation
and
Institute for High Energy Physics, NRC ”Kurchatov Institute”
Protvino 142281, Russian Federation gurevich<EMAIL_ADDRESS>
###### Abstract

We introduce analogs of creation and annihilation operators, related to
involutive and Hecke symmetries $R$, and perform bosonic and fermionic
realization of the modified Reflection Equation algebras in terms of the so-
called Quantum Doubles of Fock type. Also, we introduce Quantum Doubles of
Fock type, associated with Birman-Murakami-Wenzl symmetries coming from
orthogonal or simplectic Quantum Groups and exhibit the algebras obtained by
means of the corresponding bosonization (fermionization). Besides, we apply
this scheme to current braidings arising from Hecke symmetries $R$ via the
Baxterization procedure.

AMS Mathematics Subject Classification, 2010: 81R50

Keywords: Quantum double, creation and annihilation operators, bosonic
(fermionic) realization, half-currents, currents

##  1 Introduction

By an associative double we mean a couple $(A,B)$ of associative algebras $A$
and $B$ endowed with the so-called permutation map

$\sigma:A\otimes B\to B\otimes A,$ (1.1)

which satisfies certain requirements (see Section 3).

By a quantum double (QD) we mean an associative double where the map $\sigma$
is defined by means of a braiding — constant or current (i.e. depending on
spectral parameters), different from a (super-)flip. A typical example is the
so-called Heisenberg double constructed of an RTT algebra and a reflection
equation (RE) one, associated with the same braiding $R$. A number of other
examples of QD can be found in [GPS2].

By a bosonic QD of Fock type we mean a double composed of the algebras
$\mathrm{Sym}_{R}(V)$ and $\mathrm{Sym}_{R}(V^{*})$, where $R:V^{\otimes 2}\to
V^{\otimes 2}$ is a braiding, while $\mathrm{Sym}_{R}(V)$ and
$\mathrm{Sym}_{R}(V^{*})$ are $R$-analogs of the symmetric algebras of a
vector space $V$ and of its dual $V^{*}$. Let us precise that by a braiding
$R:V^{\otimes 2}\to V^{\otimes 2}$ we mean a solution of the braid relation

$R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23},$

where $R_{12}=(R\otimes I)$, $R_{23}=(I\otimes R)$ and $I$ stands for the
identity operator.

A fermionic version of a QD of Fock type can be defined in a similar way by
using $R$-analogs $\Lambda_{R}(V)$ and $\Lambda_{R}(V^{*})$ of the skew-
symmetric algebras $\Lambda(V)$ and $\Lambda(V^{*})$.

In the classical case, i.e. when $R$ is the usual flip $P$, the algebra
$\mathrm{Sym}(V)$ often plays the role of a Fock space. Consequently, its
generators give rise to creation operators, whereas generators of the algebra
$\mathrm{Sym}(V^{*})$ give rise to annihilation ones. The permutation
relations between the generators of these two algebras allows one to define an
action of the annihilation operators on the algebra $\mathrm{Sym}(V)$ as
partial derivatives. These two algebras form a classical version of a Fock
double. By combining the annihilation and creation operators in a proper way
we can perform a bosonic realization (or simply, a bosonization) of the Lie
algebra $gl(N)$ and consequently its enveloping algebra $U(gl(N))$.

In the present paper we construct $R$-analogs of these well-known objects,
provided braidings $R$ come from the quantum groups (QG) $U_{q}({{g}})$,
corresponding to the classical simple Lie algebras ${{g}}$. More precisely,
the role of such a braiding $R$ is played by the composition of the image of
the universal $R$-matrix in $V^{\otimes 2}$ and the usual flip (here $V$ is
the first fundamental space, called basic).

If ${{g}}\in A_{n}$, then $R$ is a Hecke symmetry. Recall that by a Hecke
symmetry we mean a braiding $R$ satisfying an additional relation

$(R-q\,I)(R+q^{-1}I)=0,\quad q\not=\pm 1.$

If ${{g}}$ belongs to other series of classical Lie algebras, then $R$ is a
Birman-Murakami-Wenzl symmetry (see Section 4). For all corresponding QG
$U_{q}({{g}})$ the $R$-symmetric algebra $\mathrm{Sym}_{R}(V)$ and $R$-skew-
symmetric algebras $\Lambda_{R}(V)$ of the space $V$ are well defined. Similar
algebras for the dual space $V^{*}$ are also well defined. Hopefully, all
these algebras are deformations of their classical counterparts (see Remark
6). The first aim of the present paper is to construct the QD from the above
algebras.

The second aim consists in studying the following problem: which algebras can
be constructed by means of the bosonization (or the fermionization) procedure
in a way similar to the classical one. It should be emphasized that the final
algebras are not the corresponding Quantum Groups themselves but they are
covariant with respect to their actions111In this connection we want to remark
that there are numerous papers (we only mention the pioneering paper [M]),
where the QG are realized via the $q$-counterparts of harmonic oscillators..
Moreover, the final algebras differ drastically from each other for different
classical series. If $R$ comes from the QG $U_{q}(sl(N))$ (in what follows we
call it the standard symmetry), we get the so-called modified reflection
equation algebra ${\cal{L}}(R)$, which is generated by the unit222All algebras
below are assumed to be unital. and elements $l_{i}^{j}$, subject to the
following system of relations:

$R_{12}L_{1}R_{12}L_{1}-L_{1}R_{12}L_{1}R_{12}=R_{12}L_{1}-L_{1}R_{12},\qquad
L=\|l_{i}^{j}\|_{1\leq i,j\leq N},\quad L_{1}=L\otimes I.$ (1.2)

This algebra is a deformation of the universal enveloping algebra $U(gl(N))$.
Observe that the algebra ${\cal{L}}(R)$ itself can be treated as an enveloping
algebra of a generalized (or braided) Lie algebra defined in the space ${\rm
End}(V)$ of internal endomorphisms. Moreover, a similar algebra ${\cal{L}}(R)$
can be associated with any Hecke symmertry $R$ (or with an involutive
symmetry, i.e. such that $R^{2}=I$), provided $R$ is skew-invertible (see the
next section for definition). A big family of such involutive and Hecke
symmetries has been constructed in [G]. For each of these symmetries the
corresponding algebra ${\cal{L}}(R)$ can be constructed by means of the
bosonization or the fermionization procedures and can be given a meaning of
the enveloping algebra of a generalized Lie algebra. Moreover, ${\cal{L}}(R)$
can be endowed with the trace $Tr_{R}$, which is coordinated with the
mentioned generalized Lie algebra structure.

Especially we are interested in the following question: to what extent the
above construction, associated with Hecke symmetries $R$, can be generalised
to the case, when $R$ is a BMW symmetry coming from the orthogonal or
simplectic QG. In this case we construct some QD associated with such
symmetries and introduce algebras arising from the bosonization or
fermionization procedure. It turns out that the final algebras differ from
${\cal{L}}(R)$ and do not have any meaning of an enveloping algebras. This
result shows that the BMW symmetries are not well adapted to constructing Lie-
algebraic objects. By contrast, they fit well for introducing group-like
objects. We refer the reader to the end of Section 4, where we discuss this
phenomenon.

Besides, we apply a similar scheme to the case when $R$ is a current braiding
constructed by means of the so-called Baxterization procedure from an
involutive or Hecke symmetry. We construct the QD of Fock type in the spirit
of the Zamolodchikov-Faddeev algebras and exhibit the algebra arising from the
corresponding bosonic realization. It turns out that this algebra is
inhomogeneous counterpart of the braided Yangians, introduced in [GS1].

The paper is organized as follows. In section 2 we recall some basic notions
and constructions, related to braidings. In section 3 we introduce the
corresponding QD of Fock type and perform the bosonic and fermionic
realizations of the algebras ${\cal{L}}(R)$. In section 4 we consider the
algebras, associated with BMW symmetries coming from the QG of the $B_{n}$,
$C_{n}$ and $D_{n}$ series. In section 5 we exhibit QD of
Zamolodchikov–Faddeev type as well as the corresponding algebras obtained by
the bosonization procedure.

Acknowledgements The work of P.S. was partially supported by RFBR grant no.
19-01-00726.

##  2 Hecke symmetries and corresponding algebras

Let $R:V^{\otimes 2}\to V^{\otimes 2}$ be a braiding. By a successive
application of $R$ it is possible to transpose arbitrary tensor powers
$V^{\otimes k}$ and $V^{\otimes l}$, that is to define a linear map

$V^{\otimes k}\otimes V^{\otimes l}\to V^{\otimes l}\otimes V^{\otimes
k},\qquad k,l\in{N}.$

However, we also need a transposition rule of the space $V$ and its dual
$V^{*}$. For this purpose we should extend the action of the operator $R$ on
spaces $V\otimes V^{*}$ and $V^{*}\otimes V$. A method of defining such an
extension belongs to V.Lyubashenko [L1, L2].

We fix a basis $\\{x_{i}\\}_{1\leq i\leq N}$ of the space $V$ and represent
the operator $R$ in the corresponding basis of the space $V^{\otimes 2}$ by an
$N^{2}\times N^{2}$ matrix $\|R_{ij}^{kl}\|$:

$R(x_{i}\otimes x_{j})=R_{ij}^{kl}\,x_{k}\otimes x_{l}.$

A summation over the repeated indices is always uderstood.

Let us fix a nondegenerate bilinear form $<\,,\,>_{r}:\,V\otimes
V^{*}\rightarrow{{C}}$ and choose a right dual basis333The term “right”
reflects the position of the argument $x^{j}$ in the bilinear form
$<\,,\,>_{r}$. $\\{x^{i}\\}_{1\leq i\leq N}$ of $V^{*}$ with respect to this
form: $<x_{i},x^{j}>_{r}=\delta_{i}^{j}$. The pairing of the spaces
$V^{\otimes 2}$ and $(V^{*})^{\otimes 2}$ is defined by

$<x_{i}\otimes x_{j},x^{k}\otimes
x^{l}>_{r}=<x_{i},x^{l}>_{r}<x_{j},x^{k}>_{r}.$

The operator $R$ on the space $(V^{*})^{\otimes 2}$ is defined as an operator
adjoint to $R$, and its action on the right dual basis vectors is given by the
rule:

$R(x^{i}\otimes x^{j})=R^{ji}_{lk}\,x^{k}\otimes x^{l}.$

We keep the notation $R$ for this adjoint operator.

A braiding $R$ is called skew-invertible if there exists an operator
$\Psi:V^{\otimes 2}\to V^{\otimes 2}$ such that

$\mathrm{Tr}_{2}R_{12}\Psi_{23}=P_{13}\quad\Leftrightarrow\quad
R_{ij}^{kl}\,\Psi_{lm}^{jn}=\delta_{m}^{k}\,\delta_{i}^{n}.$

As usual, the bottom indexes in the notation like $\Psi_{23}$ indicate the
position of components of a tensor product where the operator acts. The
aforementioned extension of the braiding $R$ on the spaces $V\otimes V^{*}$
and $V^{*}\otimes V$ reads (see [GPS1] for detail):

$R(x_{i}\otimes x^{j})=x^{k}\otimes x_{l}(R^{-1})^{lj}_{ki},\qquad
R(x^{i}\otimes x_{j})=x_{l}\otimes x^{k}\,\Psi^{li}_{kj}.$

If $R$ is skew-invertible we can define the following operators (matrices)

$B=\mathrm{Tr}_{1}\Psi_{12}\,\,\Leftrightarrow\,\,B_{i}^{j}=\Psi_{ki}^{kj},\qquad
C=\mathrm{Tr}_{2}\Psi_{12}\,\,\Leftrightarrow\,\,C_{i}^{j}=\Psi_{ik}^{jk}.$
(2.1)

If the matrices $B$ and $C$ are nonsingular, the symmetry $R$ is called
strictly skew-invertible.

Introduce now the second bilinear form $<\,,\,>_{l}:V^{*}\otimes
V\rightarrow{{C}}$ as the composition of the linear maps:

$<\,,\,>_{l}\,=\,<\,,\,>_{r}\circ R.$

Then we can introduce a left dual basis $\\{\tilde{x}^{i}\\}_{1\leq i\leq N}$
of $V^{*}$ by the requirement

$<\tilde{x}^{i},x_{j}>_{l}=\delta_{j}^{i},\quad 1\leq\forall\,i,j\leq N.$

If $R$ is strictly skew-invertible, then it is not difficult to establish the
following relations

$<x^{j},x_{i}>_{l}\,=\,B_{i}^{j},\qquad<x_{i},\tilde{x}^{j}>_{r}\,=\,(B^{-1})_{i}^{j}.$
(2.2)

Note that in the standard case the pairings $<\,,\,>_{r}$ and $<\,,\,>_{l}$
are $U_{q}(sl(N))$-covariant.

Now, consider the space of the internal endomorphisms

${\rm End}(V)\cong V\otimes V^{*}$ (2.3)

equipped with the basis $l_{i}^{j}=x_{i}\,x^{j}$. Hereafter, we omit the sign
$\otimes$, if it does not lead to misunderstanding.

By using the identification (2.3), we get the following multiplication table
in this basis

$l_{i}^{j}\circ l_{k}^{m}=B_{k}^{j}\,l_{i}^{m}.$

By contrast with the matrix $B$, determining the product $\circ$, the matrix
$C$ enters the definition of the so-called $R$-trace. Namely, for any $N\times
N$ matrix $M$ (even with noncommutative entries) its $R$-trace is defined by
the following formula $\mathrm{Tr}_{R}M=\mathrm{Tr}\,CM$.

Now, consider the following map

${\cal R}:{\rm End}(V)^{\otimes 2}\to{\rm End}(V)^{\otimes 2},\qquad
R_{12}L_{1}R_{12}L_{1}\stackrel{{\scriptstyle{\cal
R}}}{{\mapsto}}L_{1}R_{12}L_{1}R_{12},$ (2.4)

where $L=\|l_{i}^{j}\|_{1\leq i,j\leq N}$ and $L_{1}=L\otimes I$. A more
explicit expression of this map can be found in [GPS1].

Using the map ${\cal R}$, we can define an analog of the Lie bracket in the
space ${\rm End}(V)$ by putting

$[X,Y]_{R}=X\circ Y-\circ{\cal R}(X\otimes Y),\qquad X,Y\in{\rm End}(V).$
(2.5)

The data $({\rm End}(V),{\cal R},[\,\,,\,]_{R})$ is called a generalized (or
braided) Lie algebra and is denoted $gl(V_{R})$.

Observe that for this $gl(V_{R})$-bracket the following analog of the Jacobi
identity is valid on ${\rm End}(V)^{\otimes 3}$ (here we omit the subscript
$R$)

$[\,,\,]\,[\,,\,]_{23}(I-{\cal R})_{23}=[\,,\,]\,[\,,\,]_{12}.$

A proof of this relation is given in [GPS1].

Note that if $R$ is an involutive symmetry, the corresponding Jacobi identity
can be cast in the form similar to the classical one:

$[\,,\,][\,,\,]_{23}(I+{\cal R}_{12}\,{\cal R}_{23}+{\cal R}_{23}\,{\cal
R}_{12})=0.$

Let us explain the reason why the algebra ${\cal{L}}(R)$ plays the role of the
enveloping algebra of the braided Lie algebra $gl(V_{R})$. Indeed, the
defining system of the algebra ${\cal{L}}(R)$ can be cast under the following
form

$l_{i}^{j}\,l_{k}^{l}-{\cal
R}(l_{i}^{j}\,l_{k}^{l})=[l_{i}^{j},\,l_{k}^{l}]_{R},$ (2.6)

where ${\cal R}$ is the operator introduced in (2.4). Thus, by applying the
product $\circ$ to the left hand side of the relation (2.6), where we treat
the generators $l_{i}^{j}$ as elements of ${\rm End}(V)$, we get the element
$[l_{i}^{j},\,l_{k}^{l}]_{R}$.

Now, we want to show that the $R$-trace $\rm Tr_{R}$ is coordinated with the
bracket $[\,,\,]_{R}$ in the following sense

$\mathrm{Tr}_{R}\,[X,Y]_{R}=0,\quad\forall\,X,Y\in{\rm End}(V),$

which is similar to the classical relation for the trace of the commutator of
two operators. To this end we first compute ${\rm Tr}_{R}\,l_{i}^{j}$. Since
$l_{i}^{j}\triangleright x_{k}=B_{k}^{j}\,x_{i}$, then applying the $R$-trace
to the image of the element $l_{i}^{j}$ we get ${\rm Tr}_{R}\,l_{i}^{j}={\rm
Tr}\,B_{i}^{k}\,C_{k}^{j}={\alpha}\,\delta_{i}^{j}$, where ${\alpha}$ is a
nontrivial numerical factor, whose value is determined by the bi-rank of $R$
(see [GPS1] for detail). Thus, taking $X=l_{i}^{j}$ and $Y=l_{k}^{m}$, we have
to show ${\rm Tr}_{R}\,[l_{i}^{j},l_{k}^{m}]_{R}=0$. This equality is a
consequence of the following chain of relations:

${\rm Tr}_{R}\circ(R_{12}L_{1}R_{12}L_{1}-L_{1}R_{12}L_{1}R_{12})={\rm
Tr}_{R}(R_{12}L_{1}-L_{1}R_{12})={\alpha}(R_{12}I-I\,R_{12})=0.$

With any Hecke symmetry $R$ we associate the $R$-symmetric
$\mathrm{Sym}_{R}(V)$ and $R$-skew-symmetric $\Lambda_{R}(V)$ algebras of the
spaces $V$ and $V^{*}$ by setting

$\mathrm{Sym}_{R}(V)=T(V)/\langle\mathrm{Im}(q\,I-R)\rangle=T(V)/\langle
q\,x_{i}\,x_{j}-R_{i\,j}^{k\,l}\,x_{k}\,x_{l}\rangle,$ (2.7)
$\Lambda_{R}(V)=T(V)/\langle\mathrm{Im}(q^{-1}\,I+R)\rangle=T(V)/\langle
q^{-1}x_{i}\,x_{j}+R_{i\,j}^{k\,l}\,x_{k}\,x_{l}\rangle,$
$\mathrm{Sym}_{R}(V^{*})=T(V^{*})/\langle\mathrm{Im}(q\,I-R)\rangle=T(V^{*})/\langle
q\,x^{j}\,x^{i}-R^{i\,j}_{k\,l}\,x^{l}\,x^{k}\rangle,$ (2.8)
$\Lambda_{R}(V^{*})=T(V^{*})/\langle\mathrm{Im}(q^{-1}\,I+R)\rangle=T(V^{*})/\langle
q^{-1}\,x^{j}\,x^{i}+R^{i\,j}_{k\,l}\,x^{l}\,x^{k}\rangle,$

where

$T(V)=\bigoplus_{k}V^{\otimes k},\quad T(V^{*})=\bigoplus_{k}V^{*\,\otimes k}$

are the free tensor algebras and $\langle J\,\rangle$ stands for a two-sided
ideal generated by a subset $J$ in the given algebra. If $R$ is an involutive
symmetry, we put $q=1$ in all these formulae.

It should be emphasized that all these algebras are deformations of their
classical counterparts, provided $R$ is a deformation of the usual flip $P$,
i.e. dimensions of their homogenous components are classical (for a generic
$q$ if $R$ is Hecke). A similar statement is valid, if a Hecke symmetry is a
deformation of another involutive symmetry, for instance, a super-symmetry.

To complete this section, we also mention two quantum matrix algebras,
associated with any braiding $R$. One of them is the so-called RTT algebra,
defined by the system

$R_{12}T_{1}\,T_{2}=T_{1}\,T_{2}R_{12},\qquad T=\|t_{i}^{j}\|_{1\leq i,j\leq
N},\quad T_{1}=T\otimes I,\quad T_{2}=I\otimes T.$ (2.9)

The other one is the RE algebra defined via (1.2) but with the vanishing right
hand side. We do not use the former algebra. The latter one appears only under
its modified form. However, it is worth noting that if $R$ is a Hecke
symmetry, the RE algebra and its modified version are isomorphic to each
other.

##  3 Quantum doubles and bosonic (fermionic) realization of ${\cal{L}}(R)$

Our next aim is to construct some quantum doubles (QD) from the above
$R$-symmetric and $R$-skew-symmetric algebras and use them in order to perform
bosonic and fermionic realizations of the algebra ${\cal{L}}(R)$.

First, we recall the definition of doubles of associative algebras from [GS3].
Consider two associative unital algebras $A$ and $B$ equipped with a linear
map

$\sigma:A\otimes B\to B\otimes A,$

which satisfies the following conditions:

$\sigma\circ(\mu_{A}\otimes\rm{id})=(\rm{id}\otimes\mu_{A})\circ\sigma_{12}\sigma_{23}\quad{\rm
on}\,\,A\otimes A\otimes B,$
$\sigma\circ(\rm{id}\otimes\mu_{B})=(\mu_{B}\otimes\rm{id})\circ\sigma_{23}\sigma_{12}\quad{\rm
on}\,\,A\otimes B\otimes B,$ $\sigma(1_{A}\otimes b)=b\otimes
1_{A},\quad\sigma(a\otimes 1_{B})=1_{B}\otimes a,\quad\forall\,a\in
A,\,\forall\,b\in B,$

where $\mu_{A}:A\otimes A\to A$ is the multiplication operation in the algebra
$A$, $1_{A}$ is the unit element of $A$ and the symbols $\mu_{B}$ and $1_{B}$
have the same meaning in the algebra $B$. The symbol “$\rm{id}$” stands for
the identity operator.

It is not difficult to see that the bilinear map $*:(B\otimes A)^{\otimes
2}\rightarrow B\otimes A$ defined by the rule

$(b\otimes a)*(b^{\prime}\otimes
a^{\prime}):=(\mu_{B}\otimes\mu_{A})\circ(\mathrm{id}_{B}\otimes\sigma_{23}\otimes\mathrm{id}_{A})(b\otimes
a\otimes b^{\prime}\otimes a^{\prime})$

endows the space $B\otimes A$ with the structure of a unital associative
algebra and its unit element is $1_{B}\otimes 1_{A}$.

###### Definition 1

We call the data $(A,\,B,\,\sigma)$ and the corresponding algebra $(B\otimes
A,*)$ a double of associative algebras and denote it $(A,B)$.

An associative double $(A,B)$ is called quantum if the map $\sigma$ is
constructed with the use of a braiding, different from a (super-)flip.

Observe that if the both algebras $A$ and $B$ are introduced via some systems
of relations on the generators, the above conditions on the data
$(A,\,B,\,\sigma)$ mean that the permutation map preserves these relations. In
this sense we say that the permutation map and the defining relations of the
algebras $A$ and $B$ are compatible.

Now, assume the algebra $A$ to be equipped with a counit (an algebra
homomorphism)444In general we do not consider any coalgebraic structure in the
algebra $A$. So, the counit is only coordinated with the algebraic structure.
$\varepsilon_{A}:A\to{{C}}$. Then it is possible to define a linear action of
the algebra $A$ on $B$ by setting

$a\triangleright b=(\mathrm{id}_{B}\otimes\varepsilon_{A})\circ\sigma(a\otimes
b),\quad\forall\,a\in A,\,\forall\,b\in B.$

Identifying $b\otimes 1_{{C}}$ and $b$, we get that $a\triangleright b\in B$.
Thus, each element $a\in A$ defines a linear operator

$Op(a):B\to B.$

It is easy to check that the map $a\mapsto Op(a)$ defines a representation of
the algebra $A$ in the algebra $B$:

$Op(ab)=Op(a)Op(b),\quad Op(1_{A})=\mathrm{id}_{B}.$

Besides, the aforementioned Heisenberg double, we can also mention QD composed
from two copies of the RTT algebras. The permutation relations between them
are also similar to the system (2.9). A counit $\varepsilon$ defined on one
component of such a double is usually defined by
$\varepsilon(t_{i}^{j})=\delta_{i}^{j}$. Note that the corresponding
representation of the component $A$, defined according to the above scheme, is
a deformation of the trivial one $\pi(T)=I$ and consequently is not
interesting indeed. The QD of Fock type, which we are dealing with, are more
meaningful. We introduce these doubles, assuming $R$ to be a Hecke symmetry.

Let $R:V^{\otimes 2}\to V^{\otimes 2}$ be a skew-invertible Hecke symmetry,
$A=\mathrm{Sym}_{R}(V^{*})$ and $B=\mathrm{Sym}_{R}(V)$. Construct a quantum
double $(A,B)$ by introducing the following permutation relations555By
permutation relations we mean the equality of the form $a\otimes
b=\sigma(a\otimes b)$.:

$x^{a}R_{a\,i}^{b\,j}\,x_{b}=q^{-1}\,x_{i}\,x^{j}+\delta_{i}^{j}\quad\Leftrightarrow\quad
x^{l}\,x_{k}=q^{-1}\,x_{i}\,x^{j}\,\Psi_{j\,k}^{\,i\,l}+B_{k}^{l}.$ (3.1)

###### Remark 2

Recall, that we use the right dual basis in the space $V^{*}$. For the left
dual basis $\\{\tilde{x}^{j}\\}_{1\leq j\leq N}$, the permutation relations
read

$x_{b}\,R_{i\,a}^{j\,b}\,\tilde{x}^{a}=q^{-1}\,\tilde{x}^{j}\,x_{i}+\delta_{i}^{j}\quad\Leftrightarrow\quad
x_{k}\,\tilde{x}^{l}=q^{-1}\,\tilde{x}^{j}\,x_{i}\,\Psi_{k\,j}^{\,l\,i}+C_{k}^{l}.$

###### Proposition 3

The defining systems of the algebras $\mathrm{Sym}_{R}(V)$ and
$\mathrm{Sym}_{R}(V^{*})$ are compatible with the permutation relations (3.1).

Proof. Below, we use the Dirac’s “bra” and “ket” notation:

$x_{i}\rightarrow x_{|1\rangle},\quad x_{i}x_{j}\rightarrow
x_{|1\rangle}x_{|2\rangle},\quad x^{i}\rightarrow x^{\langle 1|},\quad
x^{i}x^{j}\rightarrow x^{\langle 1|}x^{\langle 2|},\,\,\dots$

We call this form of notation the matrix one. Some typical examples are

$x^{\langle 1|}x_{|1\rangle}=\sum_{i=1}^{N}x^{i}x_{i},\qquad
x_{|1\rangle}x^{\langle 1|}=x_{i}x^{j},\qquad x^{\langle
2|}R_{|12\rangle}^{\,\langle
12|}x_{|2\rangle}=\sum_{a,b=1}^{N}x^{a}R_{\,i\,a}^{\,\,j\,b}x_{b}.$

In what follows the notation $R_{|12\rangle}^{\,\langle 12|}$ will be
simplified to $R_{12}$.

Thus, the defining relations of the algebras (2.7), (2.8) and the permutation
relations (3.1) can be respectively rewritten as:

$\begin{array}[]{lcl}R_{\,i\,j}^{\,\,k\,l}\,x_{k}\,x_{l}=q\,x_{i}\,x_{j}&\quad\rightarrow&R_{12}x_{|1\rangle}x_{|2\rangle}=q\,x_{|1\rangle}x_{|2\rangle}\\\
\rule{0.0pt}{17.07164pt}x^{l}\,x^{k}R^{\,\,i\,j}_{\,k\,l}=q\,x^{j}\,x^{i}&\rightarrow&x^{\langle
2|}x^{\langle 1|}R_{12}=q\,x^{\langle 2|}x^{\langle 1|}\\\
\rule{0.0pt}{17.07164pt}x^{a}R_{\,a\,i}^{\,\,b\,j}\,x_{b}=q^{-1}\,x_{i}\,x^{j}+\delta_{i}^{j}&\rightarrow&x^{\langle
1|}R_{12}x_{|1\rangle}=q^{-1}\,x_{|2\rangle}x^{\langle
2|}+I_{|2\rangle}^{\,\langle 2|}\\\
\rule{0.0pt}{17.07164pt}&\mathrm{or}&\,x_{|2\rangle}x^{\langle
2|}=q\,x^{\langle 1|}R_{12}x_{|1\rangle}-q\,I_{|2\rangle}^{\,\langle
2|}.\end{array}$

Let us verify, for example, that the defining relations of the algebra
$\mathrm{Sym}_{R}(V)$ (2.7) are not destroyed by the permutation with vectors
from $V^{*}$. For this purpose we permute an arbitrary basis vector $x^{i}$
with elements of the generating set of the ideal in (2.7):

$\displaystyle(R_{23}\\!\\!\\!\\!$ $\displaystyle-$
$\displaystyle\\!\\!\\!\\!qI_{23})x_{|2\rangle}x_{|3\rangle}x^{\langle
3|}=(R_{23}-qI_{23})x_{|2\rangle}\left(q\,x^{\langle
2|}R_{23}x_{|2\rangle}-q\,I_{|3\rangle}^{\,\langle 3|}\right)$
$\displaystyle=$
$\displaystyle\\!\\!\\!-q(R_{23}-qI_{23})x_{|2\rangle}+q(R_{23}-qI_{23})\Big{(}qx^{\langle
1|}R_{12}x_{|1\rangle}-qI_{|2\rangle}^{\langle 2|}\Big{)}R_{23}x_{|2\rangle}$
$\displaystyle=$ $\displaystyle q^{2}x^{\langle
1|}(R_{23}-qI_{23})R_{12}R_{23}x_{|1\rangle}x_{|2\rangle}=q^{2}x^{\langle
1|}R_{12}R_{23}(R_{12}-qI_{12})x_{|1\rangle}x_{|2\rangle}.$

Here, by passing to the third line we have taken into account the consequence
of the Hecke condition $(R_{23}-qI_{23})R_{23}=-q^{-1}(R_{23}-qI_{23})$, while
the last equality is due to the braid relation for $R$.

So, we finally get the equality:

$(R_{23}-qI_{23})x_{|2\rangle}x_{|3\rangle}\,x^{\langle 3|}=q^{2}x^{\langle
1|}R_{12}R_{23}\,(R_{12}-qI_{12})x_{|1\rangle}x_{|2\rangle},$

which means the compatibility of the defining relations of the algebra
$\mathrm{Sym}_{R}(V)$ with the permutation relations (3.1).

The compatibility condition for the defining relations of
$\mathrm{Sym}_{R}(V^{*})$ and the permutation relations can be verified in the
same way.

Now, we introduce the counit similarly to the classical case:

$\varepsilon_{A}:A=\mathrm{Sym}_{R}(V^{*})\to{{C}},\qquad\varepsilon_{A}(1_{A})=1_{{{C}}},\quad\varepsilon_{A}(x^{j})=0,\quad
1\leq\forall j\leq N.$

According to the above scheme all elements of the algebra
$A=\mathrm{Sym}_{R}(V^{*})$ can be represented by operators acting on the
algebra $B=\mathrm{Sym}_{R}(V)$. Compute the action of elements $x^{j}$ onto
some monomials in $x_{i}$:

$x^{j}\triangleright 1_{B}=0,\quad x^{j}\triangleright x_{i}=B_{i}^{j},\quad
x^{j}\triangleright(x_{i}\,x_{k})=B_{i}^{j}\,x_{k}+q^{-1}\,\,B^{l}_{k}\Psi_{li}^{mj}\,x_{m},\quad{\rm
etc}.$ (3.2)

Note that using the properties of $\Psi$ and $R$ we can rewrite the above
actions in the equivalent matix form:

$x^{\langle
1|}\triangleright\Big{(}R_{12}x_{|1\rangle}\Big{)}=I_{|2\rangle}^{\,\langle
2|},\qquad x^{\langle
1|}\triangleright\Big{(}R_{12}R_{23}x_{|1\rangle}x_{|2\rangle}\Big{)}=(R_{23}+q^{-1}I_{23})x_{|2\rangle}.$

As we noticed above, the generators of the algebra $A=\mathrm{Sym}_{R}(V^{*})$
are analogs of the usual annihilation operators, whereas the generators of
$B=\mathrm{Sym}_{R}(V)$ are analogs of the creation operators: the result of
the action of $x_{i}$ on $a\in\mathrm{Sym}_{R}(V)$ is the product $x_{i}a$.
The above double $(A,B)$ is said to be quantum bosonic double of Fock type.

In a similar manner it is possible to construct a quantum fermionic double of
Fock type by setting $A={\mathrm{\Lambda}}_{R}(V^{*})$,
$B={\mathrm{\Lambda}}_{R}(V)$ and

$x^{a}R_{a\,i}^{\,b\,j}\,x_{b}=-q\,x_{i}\,x^{j}+\delta_{i}^{j}\quad\Leftrightarrow\quad
x^{l}\,x_{k}=-q\,x_{i}\,x^{j}\,\Psi_{j\,k}^{\,i\,l}+B_{k}^{l}.$ (3.3)

In this case all defining relations are also compatible.

Now, compose the matrix $L=\|l_{i}^{j}\|$, $l_{i}^{j}=x_{i}x^{j}$, where
$x_{i}\in B$, $x^{j}\in A$ and $(A,B)$ is a bosonic or fermionic double of
Fock type. Then the following proposition is valid.

###### Proposition 4

The matrix $L$ meets the relations (1.2).

Proof. In the bosonic case this proposition was proved in [GS1]. In the
fermionic case the proof is similar.

Thus, representing the factors of the product $x_{i}x^{j}$ as the creation and
annihilation operators acting on the algebra $B$, we get the so-called bosonic
(or fermionic) realization of the algebra ${\cal{L}}(R)$ similar to that of
the algebra666Note that if $R\to P$ as $q\to 1$ (for instance, if $R$ is a
standard symmetry) the system (1.2) turns into the matrix equality
$PL_{1}PL_{1}-L_{1}PL_{1}P=PL_{1}-L_{1}P\quad\Leftrightarrow\quad
l_{i}^{j}\,l_{k}^{m}-l_{k}^{m}\,l_{i}^{j}=l_{i}^{m}\,\delta_{k}^{j}-l_{k}^{j}\,\delta_{i}^{m},$
valid for the usual generators of the algebra $U(gl(N))$. $U(gl(N))$.

Moreover, similarly to the classical case the generators $l_{i}^{j}$ preserve
any homogenous component $\mathrm{Sym}_{R}^{k}(V)$, $k\geq 1$. Thus, we get a
series of finite-dimensional representations of the algebra ${\cal{L}}(R)$.
For instance, the action of the elements $l_{i}^{j}=x_{i}x^{j}$ onto the first
component $\mathrm{Sym}_{R}^{1}(V)=V$ is as follows

$l_{i}^{j}\triangleright x_{k}=x_{i}\triangleright(x^{j}\triangleright
x_{k})=B^{j}_{k}\,x_{i}.$ (3.4)

###### Remark 5

Note that on the role of the algebra $B$ we can assign the free tensor algebra
$T(V)$ instead of $\mathrm{Sym}_{R}(V)$ and keep the same permutation
relations. By considering all invariant subspaces with respect to the action
$\triangleright$, it is possible to construct a more rich representation
category of the algebra ${\cal{L}}(R)$. Such a category was constructed [GPS1]
by means of a “braided bi-algebra structure” of ${\cal{L}}(R)$. Our present
method based on bosonization is a sense close to the method of constructing
differential calculus on the quantum hyperplane from [WZ, H].

##  4 Quantum doubles associated with BMW symmetries

Let us recall that by a BMW symmetry we mean a braiding $R$ which comes from a
QG belonging to one of the orthogonal ($B_{n},\,D_{n}$) or simplectic
($C_{n}$) series. An explicit form of these symmetries are exhibited in [FRT].
Each of them is subject to some relations. We need only one of them:

$(R-qI)(R+q^{-1}I)(R-\mu I)=0.$ (4.1)

Note that if $R$ comes from the QG corresponding to an orthogonal group then
$\mu=q^{1-N}$ and if $R$ comes from the QG corresponding to a simplectic
group, then $\mu=-q^{-1-N}$. Here $N$ is the dimension of the basic vector
space. Below, for the sake of simplicity we speak about orthogonal and
simplectic QG.

Also, there are known examples of the BMW symmetries coming from super-QG with
other values of $\mu$ (see [I]). Below, we only deal with the BMW symmetries,
coming from orthogonal or simplectic QG. Consequently, the parameter $\mu$ is
assumed to take one of the above values. Thus, as $q\to 1$, this parameter
tends to 1 for the orthogonal QG and to -1 for the simplectic ones.

In [OP] there were considered the so-called BMW algebras, which are defined by
similar relations but with the parameter $\mu$ independent on $q$.

In virtue of (4.1) there exist three complementary idempotents ${\cal P}^{q}$,
${\cal P}^{-q^{-1}}$ and ${\cal P}^{\mu}$ such that

$R=q\,{\cal P}^{q}-q^{-1}{\cal P}^{-q^{-1}}+\mu{\cal P}^{\mu}.$

Each of them is a projector onto the corresponding eigenspace and can be
explicitly written in terms of $R$. For instance, the projector

${\cal P}^{-q^{-1}}=\frac{(R-qI)(R-\mu I)}{(q+q^{-1})(q^{-1}+\mu)}$ (4.2)

maps the space $V^{\otimes 2}$ onto the eigenspace of the operator $R$
corresponding to the eigenvalue $-q^{-1}$. Note that for a generic $q$ the
denominator of this formula does not vanish.

Now, let us define the $R$-symmetric and $R$-skew-symmetric algebras of the
space $V$ by setting

$\mathrm{Sym}_{R}(V)=T(V)/\langle\mathrm{Im}\,{\cal
P}^{-q^{-1}}\rangle,\qquad{\mathrm{\Lambda}}_{R}(V)=T(V)/\langle\mathrm{Ker}\,{\cal
P}^{-q^{-1}}\rangle,$ (4.3)

provided $R$ comes from an orthogonal QG and

$\mathrm{Sym}_{R}(V)=T(V)/\langle\mathrm{Ker}\,{\cal
P}^{q}\rangle,\qquad{\mathrm{\Lambda}}_{R}(V)=T(V)/\langle\mathrm{Im}\,{\cal
P}^{q}\rangle,$

provided $R$ comes from a simplectic QG.

###### Remark 6

Note that according to [FRT] $R$-symmetric algebras have the classical
Poincaré series, i.e. these series are equal to the series corresponding to
the algebras $\mathrm{Sym}(V)$. However, any prof of this claim has never been
published. We do not use this property. We want only to observe that the space
$\mathrm{Ker}\,{\cal P}^{\mu}$ belongs to the second homogenous $R$-symmetric
component of the space $V^{\otimes 2}$ if the QG is orthogonal and to
$R$-skew-symmetric component if QG is simplectic.

In a similar manner we define the $R$-symmetric and $R$-skew-symmetric
algebras of the space $V^{*}$. For instance, we put (in the orthogonal case)

$\mathrm{Sym}_{R}(V^{*})=T(V^{*})/\langle\mathrm{Im}\,{\cal
P}^{-q^{-1}}\rangle,$ (4.4)

where the idempotents ${\cal P}^{q}$, ${\cal P}^{-q^{-1}}$ and ${\cal
P}^{\mu}$ acting in the space $(V^{*})^{\otimes 2}$ are expressed in terms of
$R$ by the same formulae as above. We only have to take into account the way
of extending the braiding $R$ onto the space $V^{*\otimes 2}$, namely,
${R}(x^{k}\otimes x^{l})={R}_{ji}^{lk}(x^{i}\otimes x^{j})$.

Let us define the permutation relations by formula (3.1) for an orthogonal QG
and by (3.3) for a simplectic QG. Then the following claim is valid.

###### Proposition 7

The permutation relations are compatible with the defining systems of the
algebras $\mathrm{Sym}_{R}(V)$ and $\mathrm{Sym}_{R}(V^{*})$ for the
orthogonal series and with these of the algebras ${\mathrm{\Lambda}}_{R}(V)$
and ${\mathrm{\Lambda}}_{R}(V^{*})$ for the simplectic series.

Proof. The proposition is proved by a straightforward calculation similar to
the case of the Hecke symmetries (see Proposition 3 above). Nevertheless, we
give a short sketch of the proof in order to stress the difference with the
Hecke case.

Consider the algebra $\mathrm{Sym}_{R}(V)$ for the orthogonal QG defined by
the quotient (4.3). Taking into account the permutation relations (3.1) we get
the following transformation:

$x_{|2\rangle}x_{|3\rangle}x^{\langle 3|}=q^{2}x^{\langle
1|}R_{12}R_{23}x_{|1\rangle}x_{|2\rangle}-q^{2}(R_{23}+q^{-1}I_{23})\,x_{|2\rangle}.$

Then for the generating set of the ideal $\langle\,\mathrm{Im}\,{\cal
P}^{-q^{-1}}\rangle$ we have:

${\cal P}^{-q^{-1}}(R_{23})x_{|2\rangle}x_{|3\rangle}x^{\langle
3|}=q^{2}x^{\langle 1|}R_{12}R_{23}{\cal
P}^{-q^{-1}}(R_{12})x_{|1\rangle}x_{|2\rangle}-q^{2}{\cal
P}^{-q^{-1}}(R_{23})(R_{23}+q^{-1}I_{23})\,x_{|2\rangle}.$

The first therm in the right hand side is a consequence of the braid relation
on $R$:

${\cal P}^{-q^{-1}}(R_{23})R_{12}R_{23}=R_{12}R_{23}{\cal
P}^{-q^{-1}}(R_{12}),$

while the second term is equal to zero due to the definition (4.2) of the
projector ${\cal P}^{-q^{-1}}$ and the cubic minimal polynomial (4.1) of the
symmetry $R$. So, we finally get

$\underline{{\cal P}^{-q^{-1}}(R_{23})x_{|2\rangle}x_{|3\rangle}}x^{\langle
3|}=q^{2}x^{\langle 1|}R_{12}R_{23}\,\underline{{\cal
P}^{-q^{-1}}(R_{12})x_{|1\rangle}x_{|2\rangle}},$

which means the compatibility of the permutatuon relations (3.1) with the
defining relations of the algebra $\mathrm{Sym}_{R}(V)$ for the orthogonal
case. All other compatibilities are verified by the similar calculations.

Now, consider the QD $(A,B)$ of Fock type, corresponding to a BMW symmetry
$R$. For $R$ corresponding to the orthogonal QG we set
$A=\mathrm{Sym}_{R}(V^{*})$, $B=\mathrm{Sym}_{R}(V)$, and for $R$
corresponding to a simplectic QG we set $A={\Lambda}_{R}(V^{*})$,
$B={\Lambda}_{R}(V)$. By introducing the same counit as in the algebra
${\cal{L}}(R)$, we get an action of the algebra $A$ on the algebra $B$. The
creation and annihilation operators are similar to these from the bosonic
(resp., fermionic) realization of the algebra ${\cal{L}}(R)$.

Now, we introduce a matrix $L=\|l_{i}^{j}\|$ with entries
$l_{i}^{j}=x_{i}x^{j}$.

###### Proposition 8

For all series the matrix $L$ meets the following system

${\cal P}_{12}L_{1}R_{12}L_{1}-L_{1}R_{12}L_{1}{\cal P}_{12}={\cal
P}_{12}L_{1}-L_{1}{\cal P}_{12},$ (4.5)

where ${\cal P}={\cal P}^{q}+{\cal P}^{\mu}$ for the orthogonal series and
${\cal P}={\cal P}^{-q^{-1}}+{\cal P}^{\mu}$ for the simplectic one.

Proof. Consider the case of $R$ corresponding to an orthogonal QG in detail.
For a simplectic QG all considerations are analogous. Below, we use the matrix
notation introduced above.

Since the idempotents ${\cal P}^{q}$, ${\cal P}^{-q^{-1}}$ and ${\cal
P}^{\mu}$ are complementary, we have

${\cal P}^{-q^{-1}}+{\cal P}^{q}+{\cal P}^{\mu}=I.$

Also, according to the definitions of the algebras $\mathrm{Sym}_{R}(V)$ and
$\mathrm{Sym}_{R}(V^{*})$ and the permutation relations the complete system on
the generators $x_{i}$ and $x^{i}$ are

${\cal P}_{12}x_{|1\rangle}x_{|2\rangle}=x_{|1\rangle}x_{|2\rangle},\qquad
x^{\langle 2|}x^{\langle 1|}{\cal P}_{12}=x^{\langle 2|}x^{\langle 1|},\qquad
x^{\langle 1|}R_{12}x_{|1\rangle}=q^{-1}x_{|2\rangle}x^{\langle 2|}+I_{2},$

where ${\cal P}={\cal P}^{q}+{\cal P}^{\mu}$.

Now, substitute the matrix $L_{1}=x_{|1\rangle}x^{\langle 1|}$ into the left
hand side of (4.5). Using the relations on the generators we transform the
first summand to the following expression:

${\cal P}_{12}x_{|1\rangle}\underline{x^{\langle
1|}R_{12}x_{|1\rangle}}x^{\langle 1|}=q^{-1}\underline{{\cal
P}_{12}x_{|1\rangle}x_{|2\rangle}}x^{\langle 2|}x^{\langle 1|}+{\cal
P}_{12}x_{|1\rangle}x^{\langle 1|}=q^{-1}x_{|1\rangle}x_{|2\rangle}x^{\langle
2|}x^{\langle 1|}+{\cal P}_{12}L_{1}.$

Here, underlined are the terms which undergo changes. By the same steps of
transformations the second summand can be rewritten in the form:

$x_{|1\rangle}\underline{x^{\langle 1|}R_{12}x_{|1\rangle}}x^{\langle 1|}{\cal
P}_{12}=q^{-1}x_{|1\rangle}x_{|2\rangle}\underline{x^{\langle 2|}x^{\langle
1|}{\cal P}_{12}}+x_{|1\rangle}x^{\langle 1|}{\cal
P}_{12}=q^{-1}x_{|1\rangle}x_{|2\rangle}x^{\langle 2|}x^{\langle
1|}+L_{1}{\cal P}_{12}.$

By taking the difference of the above expressions, we get (4.5).

Observe that the algebra defined by (4.5) cannot be treated as the enveloping
algebra $U({{g}})$ of a generalized Lie algebra, since the relation (4.5) does
not enable us to define any operator $\rm{End}(V)^{\otimes
2}\to\rm{End}(V)^{\otimes 2}$ similar to ${\cal R}$777Observe that if $R$ is a
Hecke symmetry, the corresponding modified RE algebra can be cast in the form
(4.5), but with ${\cal P=\cal P}^{q}$..

Moreover, though the algebras $\mathrm{Sym}_{R}(V)$ and $\Lambda_{R}(V)$ are
well defined in this case and are (hopefully!) deformations of their classical
counterparts, the algebras obtained via the bosonic (fermionic) realization
are not deformations of $U({{g}})$.

Nevertheless, proceeding in a similar way as described above we can construct
finite dimensional representations of the algebra defined by (4.5) in
homogenous components of the algebras $\mathrm{Sym}_{R}(V)$ or
${\Lambda}_{R}(V)$ depending on the series.

Concluding this section we note that if $R$ is an involutive or Hecke
symmetry, deforming the usual flip $P$, the both algebras
$\mathrm{Fun}(GL(N))$ and $U(gl(N))$ admit quantum deformations. Moreover,
there are two different deformations of the algebra $\mathrm{Fun}(GL(N))$: the
corresponding RTT algebra or the RE one. However, a deformation of the algebra
$U(gl(N))$ can be performed only via the modified RE algebra.

If ${{g}}$ is an orthogonal or simplectic Lie algebra, the space
$\mathrm{Fun}(G)$ of functions on the corresponding group $G$ can be deformed
as was done in [OP]. Note that the authors of the cited paper use Quantum
Matrix Algebras of a general form including the RTT and RE algebras. By
contrary, the algebra $U({{g}})$ does not have any deformation, covariant with
respect to the adjoint action of the corresponding QG $U_{q}({{g}})$.

As we have noticed in Introduction, the BMW symmetries are not well adapted to
constructing $R$-analogs of Lie algebras.

###### Remark 9

Now, we would like to discuss the Poisson counterparts of the algebras
obtained by the bosonization. If $R$ is a Hecke symmetry deforming the usual
flip $P$ (for instance, that coming from the QG $U_{q}(sl(N))$ or the Crammer-
Gervais one), then the Poisson counterpart of the algebra ${\cal{L}}(R)$ (more
precisely, of the algebra, obtained by introducing a second deformation
parameter $\mathchar 1406\relax$ in the front of the right hand side of (1.2))
is a pencil defined on the commutative algebra $\mathrm{Sym}(gl(N))$. This
pencil is generated by the linear Poisson-Lie bracket, associated with the Lie
algebra $gl(N)$, and by a quadratic bracket, corresponding to the case
$\mathchar 1406\relax=0$. It is interesting to observe that almost all
brackets of this pencil are not unimodular. Nevertheless, the quantum
counterpart of this pencil (namely, the algebra ${\cal{L}}(R)$) is endowed
with a trace, but this trace is not usual.

If $R$ is a BMW symmetry coming from an orthogonal or simplectic QG, a similar
Poisson pencil does not exist. However, a Poisson bracket, corresponding to
the deformation of the algebra $\mathrm{Fun}(G)$ can be defined.

##  5 Quantum double of Zamolodchikov-Faddeev type

In this section we deal with algebras associated with current braidings. By a
current braiding we mean an operator $R(u,v)$ depending on spectral parameters
subject to the relation

$R_{12}(u,v)\,R_{23}(u,w)\,R_{12}(v,w)=R_{23}(v,w)\,R_{12}(u,w)\,R_{23}(u,v).$

The current braidings, we are dealing with, arise from the Baxterization
procedure applied to an involutive (resp., Hecke) symmetry. These current
braiding are of the form

$R(u,v)=R-\frac{I}{u-v},\quad\mathrm{resp.,}\quad
R(u,v)=R-\frac{(q-q^{-1})\,u\,I}{u-v},$ (5.1)

where $R$ is any skew-invertible involutive (resp., Hecke) symmetry. The
former braiding $R(u,v)$ is called rational, the latter one is called
trigonometric. The reader is referred to [GS1] for details.

Let us pass to the normalized braidings ${\bf{R}}(u,v)=g(u,v)^{-1}R(u,v)$,
where

$g(u,v)=1-\frac{1}{u-v},\quad\mathrm{resp.}\quad
g(u,v)=q-\frac{(q-q^{-1})u}{u-v}$

provided $R(u,v)$ is a rational (resp., trigonometrical) braiding (5.1).

It should be emphasized that these operators are still subject to the braid
relation (5.1) and are involutive in the following sense

${\bf{R}}(u,v)\,{\bf{R}}(v,u)=I.$

In virtue of this relation the operators ${\bf{R}}(u,v)$ have two eigenvalues
$\pm 1$. So, we introduce an analog of the algebra $\mathrm{Sym}_{R}(V)$ by
the following relations:

${\bf{R}}_{ij}^{kl}(u,v)x_{k}(u)\,x_{l}(v)=x_{i}(v)\,x_{j}(u)\quad\Leftrightarrow\quad
R_{ij}^{kl}(u,v)x_{k}(u)\,x_{l}(v)=g(u,v)\,x_{i}(v)\,x_{j}(u).$ (5.2)

Write this system in a more detailed form for a trigonometric braiding

$q\,x_{i}(v)\,x_{j}(u)-R_{ij}^{kl}x_{k}(u)\,x_{l}(v)=\frac{(q-q^{-1})\,u}{u-v}(x_{i}(v)\,x_{j}(u)-x_{i}(u)\,x_{j}(v)).$

(The defining relations of the algebra $\Lambda_{R}(V)$ can be written in a
similar way.)

However, if we try to represent the currents $x_{i}(u)$ as a formal series in
$u$ with Fourier coefficients $x_{i}[m]$ as follows

$x_{i}(u)=\sum_{m\in{{Z}}}x_{i}[m]\,u^{-m-1},$

we find that it is not possible to rewrite the defining system via polynomial
relations on these coefficients. By following the classical pattern, we
present the algebra in terms of the so-called half-currents

$x_{i}^{+}(u)=\sum_{m\in{{Z}},\,m<0}x_{i}[m]\,u^{-m-1},\quad
x_{i}^{-}(u)=\sum_{m\in{{Z}},\,m\geq 0}x_{i}[m]\,u^{-m-1},\quad
x_{i}(u)=x_{i}^{+}(u)+x_{i}^{-}(u).$

Then by imposing the relations

$\begin{array}[]{l}g(u,v)\,x_{i}^{\pm}(v)\,x_{j}^{\pm}(u)=R_{ij}^{kl}(u,v)x_{k}^{\pm}(u)\,x_{l}^{\pm}(v),\\\
\rule{0.0pt}{19.91692pt}g(u,v)\,x_{i}^{-}(v)\,x_{j}^{+}(u)=R_{ij}^{kl}(u,v)x_{k}^{+}(u)\,x_{l}^{-}(v),\end{array}$
(5.3)

and by using the expansion $\frac{1}{u-v}=\sum_{p\geq
0}\frac{v^{p}}{u^{p+1}}$, we can rewrite (5.2) in terms of the Fourier
coefficients. However, the system (5.2) for the currents $x_{i}(u)$ is
preserved.

According to the scheme exhibited in Section 3, we introduce the current
$R$-symmetric algebra of the dual space by the following system

$g(u,v)\,x^{i}(u)\,x^{j}(v)=R^{ji}_{lk}(u,v)x^{k}(v)\,x^{l}(u).$ (5.4)

This system can be also expressed via the Fourier coefficients if we pass to
the corresponding half-currents $x^{i}_{\pm}(u)$. Besides, we introduce the
permutation relations

$x^{a}(u)\,R_{a\,i}^{b\,j}\,x_{b}(v)=q^{-1}\,x_{i}(v)\,x^{j}(u)+\delta_{i}^{j}\,\delta(u-v),$
(5.5)

which are compatible with the relations on the currents

$x_{i}(u)=x_{i}^{+}(u)+x_{i}^{-}(u)\,\,{\rm
and}\,\,x^{j}(u)=x^{j}_{+}(u)+x^{i}_{-}(u).$

Thus, we have again constructed a quantum double $(A,B)$ of Fock type. The
algebras $A$ and $B$ are generated by the Fourier coefficients of the currents
$x^{k}(u)$ and $x_{k}(u)$, which respectively play the role of the
annihilation and creation operators. Using the definition
$\delta(u-v)=\sum_{p\in{{Z}}}\frac{v^{p}}{u^{p+1}}$, we can express the system
(5.5) in terms of the Fourier coefficients:

$x^{a}[k]R_{a\,i}^{b\,j}\,x_{b}[l]=q^{-1}\,x_{i}[l]\,x^{j}[k]+\delta_{i}^{j}\,\delta_{k+l}^{1}\quad\Leftrightarrow\quad
x^{a}[k]\,x_{b}[l]=q^{-1}\,x_{i}[l]\,x^{j}[k]\,\Psi_{j\,b}^{\,i\,a}+B_{b}^{a}\,\delta_{k+l}^{1}.$

Now, define the counit $\varepsilon_{A}$ as above by setting

$\varepsilon_{A}(1_{A})=1_{{{C}}},\quad\varepsilon_{A}(x^{a}[k])=0,\quad\forall\,a,\,k.$

This allows us to get the action of the algebra $A$ on $B$:

$x^{a}[k]\triangleright x_{b}[l]=B_{b}^{a}\delta_{k+l}^{1}\,\,{\rm
and\,\,so\,\,on}.$

We call the algebra, defined by the relations (5.2), (5.4), and (5.5), the QD
of Zamolodchikov-Faddeev type.

Now, introduce the matrix $L(u)=\|l_{i}^{j}(u)\|_{1\leq i,j,\leq N}$ where
$l_{i}^{j}(u)=x_{i}(u)\,x^{j}(u)$.

###### Proposition 10

The matrix $L(u)$ meets the following relation

$R_{12}(u,v)L_{1}(u)R_{12}L_{1}(v)-L_{1}(v)R_{12}L_{1}(u)R_{12}(u,v)=(R_{12}L_{1}(u)-L_{1}(u)R_{12})\,\delta(u-v).$
(5.6)

Proof. Again, we use the matrix notation. Consider the first summand in the
left hand side of (5.6) and substitute the matrices $L(u)$ expressed in terms
of currents $x_{i}(u)$ and $x^{j}(u)$:

$\displaystyle R_{12}(u,v)x_{|1\rangle}(u)\underline{x^{\langle
1|}(u)R_{12}x_{|1\rangle}(v)}x^{\langle
1|}(v)\stackrel{{\scriptstyle(\ref{pe})}}{{=}}q^{-1}\underline{R_{12}(u,v)x_{|1\rangle}(u)x_{|2\rangle}(v)}x^{\langle
2|}(u)x^{\langle 1|}(v)$ $\displaystyle+R_{12}(u,v)x_{|1\rangle}(u)x^{\langle
1|}(v)\,\delta(u-v)\stackrel{{\scriptstyle(\ref{chet})}}{{=}}q^{-1}g(u,v)x_{|1\rangle}(u)x_{|2\rangle}(v)x^{\langle
2|}(u)x^{\langle 1|}(v)$ $\displaystyle+R_{12}(u,v)x_{|1\rangle}(u)x^{\langle
1|}(v)\,\delta(u-v).$

In a similar way (the only difference is using (5.4) instead of (5.2)) we
transform the second summand in the left hand side of (5.6):

$L_{1}(v)R_{12}L_{1}(u)R_{12}(u,v)=q^{-1}g(u,v)x_{|1\rangle}(u)x_{|2\rangle}(v)x^{\langle
2|}(u)x^{\langle 1|}(v)+x_{|1\rangle}(v)x^{\langle
1|}(u)R_{12}(u,v)\,\delta(u-v).$

Taking the difference of the above expressions we get the desired result:

$\displaystyle
R_{12}(u,v)L_{1}(u)R_{12}L_{1}(v)-L_{1}(v)R_{12}L_{1}(u)R_{12}(u,v)=$
$\displaystyle(R_{12}(u,v)x_{|1\rangle}(u)x^{\langle
1|}(v)-x_{|1\rangle}(v)x^{\langle
1|}(u)R_{12}(u,v))\delta(u-v)=(R_{12}L_{1}(u)-L_{1}(u)R_{12})\,\delta(u-v).$

Note that the product $R(u,v)\delta(u-v)$ contains the ill-defined terms
$\frac{(q-q^{-1})u}{u-v}\,\delta(u-v)$, but they cancel each other in the
final expression. As a consequence, the right hand side of (5.6) contains only
constant matrix $R$.

Concluding the paper we want to observe that the algebras analogous to those
defined by (5.6), but with vanishing right hand side, were introduced in [GS1,
GS2] under the name of the braided Yangians or generalized Yangians of RE
type. Similar algebras but with current braidings in the middle positions and
depending of a charge were considered in [GS4]. Note that the algebras from
[GS4] are generalizations of those from [RS], corresponding to the $A_{n}$
series.

Nevertheless, the defining relations of the algebras from [GS4] cannot be
expressed via the Fourier components. By contrast, the system (5.6) can be
written in terms of these components. This is a consequence of the fact that
we define the permutation relations (5.5) in the QD above by means of a
constant braiding (namely, a Hecke symmetry), in contrast with the initial
definition of the ZF algebras.

## References

  * [FRT] Reshetikhin N., Takhtadzhyan L., Faddeev L. Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1 (1990) 193–225.
  * [G] Gurevich D. Algebraic aspects of the Yang-Baxter equation, Leningrad Math. J. 2 (1991) 801 – 828.
  * [GPS1] Gurevich D., Pyatov P., Saponov P. Representation theory of (modified) reflection equation algebra of $GL(m|n)$ type, St. Petersburg Math. J. 20 (2008) 213–253.
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