Bi-Hamiltonian Systems in (2+1) and Higher Dimensions Defined by Novikov
Algebras

Bi-Hamiltonian Systems in (2+1)
and Higher Dimensions Defined by Novikov Algebras

Błażej M. SZABLIKOWSKI

B.M. Szablikowski

Faculty of Physics, Division of Mathematical Physics, Adam Mickiewicz
University,
ul. Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland<EMAIL_ADDRESS>
Received June 21, 2019, in final form November 21, 2019; Published online
November 29, 2019

The results from the article [Strachan I.A.B., Szablikowski B.M., Stud. Appl.
Math. 133 (2014), 84–117] are extended over consideration of central
extensions allowing the introducing of additional independent variables.
Algebraic conditions associated to the first-order central extension with
respect to additional independent variables are derived. As result $(2+1)$\-
and, in principle, higher-dimensional multicomponent bi-Hamiltonian systems
are constructed. Necessary classification of the central extensions for low-
dimensional Novikov algebras is performed and the theory is illustrated by
significant $(2+1)$\- and $(3+1)$-dimensional examples.

Novikov algebras; $(2+1)$\- and $(3+1)$-dimensional integrable systems; bi-
Hamiltonian structures; central extensions

37K10; 17B80; 37K30

##  1 Introduction

In the article [21] we have presented construction of $(1+1)$-dimensional
integrable bi-Hamiltonian systems associated with Novikov algebras. These
systems are multicomponent generalizations of the Camassa–Holm equation [8]
and can be interpreted as Euler equations on the respective centrally extended
Lie algebras. On the other hand, the central extension procedure is one of the
most effective methods allowing for the construction of $(2+1)$ analogs of
$(1+1)$-dimensional systems. However, such procedure is not always possible.
All the more, there is very limited number of approaches allowing for the
systematic construction of $(3+1)$\- and higher-dimensional integrable
systems, see the recent survey [6] and references therein.

Novikov algebras naturally appear in the context of the homogeneous first-
order Hamiltonian operators [4, 12]:

$\displaystyle\mathcal{P}^{ij}=\big{(}b^{ij}_{k}+b^{ji}_{k}\big{)}u^{k}\partial_{x}+b^{ij}_{k}u^{k}_{x},$
(1.1)

which are a special case of Dubrovin–Novikov operators of hydrodynamic type
[9, 10]. The algebraic conditions for (1.1) to be Hamiltonian means that
$b^{ij}_{k}$ are structure constants of a Novikov algebra. The Hamiltonian
operators (1.1) define Lie–Poisson structures associated with the so-called
translationally invariant Lie algebras that are in one to one correspondence
with Novikov algebras. For more information about this and directly related
topics see [21] and the recent works [19, 20, 22].

Novikov algebras also play significant role in the theory of multi-dimensional
Hamiltonian operators of Dubrovin–Novikov type:

$\displaystyle\mathcal{P}^{ij}=\sum_{\alpha=1}^{D}\big{[}g^{ij\alpha}({\bm{u}})\partial_{x^{\alpha}}+b^{ij\alpha}_{k}({\bm{u}})u^{k}_{x^{\alpha}}\big{]},\qquad{\bm{u}}=\big{(}u^{1},\ldots,u^{n}\big{)},$
(1.2)

which in the non-degenerate case, $\det g^{ij\alpha}({\bm{u}})\neq 0$, can be
reduced by a change of dependent variables to a linear form,111One-dimensional
Dubrovin–Novikov operator ($D=1$) can be reduced to a constant form. where

$\displaystyle
g^{ij\alpha}({\bm{u}})=\big{(}b^{ij\alpha}_{k}+b^{ji\alpha}_{k}\big{)}u^{k}+g^{ij\alpha}_{0},\qquad
b^{ij\alpha}_{k}({\bm{u}})=b^{ij\alpha}_{k}=\text{const},\qquad
g^{ij\alpha}_{0}=\text{const},$

and $b^{ij\alpha}_{k}$ are structure constants of Novikov algebras that
additionally satisfy some compatibility conditions, see [14, 15] and also
[20].

The key result of this paper is the use of $(N+1)$-dimensional pencils

$\displaystyle\mathcal{P}^{ij}_{\lambda}=\mathcal{P}^{ij}+\mathcal{P}^{ij}_{1}+\lambda\mathcal{P}^{ij}_{0},\qquad\lambda\in\mathbb{C},$
(1.3)

that are central extension of the first-order homogeneous Hamiltonian
operators (1.1), for the construction of respective $(N+2)$-dimensional bi-
Hamiltonian integrable hierarchies. The extension in (1.3) is defined through
admissible cocycles:

$\displaystyle\mathcal{P}^{ij}_{\mu}=g^{ij}_{\mu}\partial_{x}+f^{ij}_{\mu}\partial^{2}_{x}+h^{ij}_{\mu}\partial_{x}^{3}+\sum_{\alpha=1}^{N}\eta^{ij}_{\mu,\alpha}\partial_{y^{\alpha}},\qquad\mu=0,1,$

where $N$ is the dimension of the linear space spanned by independent cocycles
introducing additional spatial variables $y^{\alpha}$. In Theorem 3.1 we
derive the algebraic conditions that must be satisfied on a Novikov algebra by
the first-order central extension defined with respect to an additional
independent variable. Further, we show that the Killing forms must practically
satisfy the same algebraic conditions. Subsequently, we classify such central
extensions with respect to low-dimensional Novikov algebras, that have been
originally classified in the articles [2, 3, 7] and those that can be
extracted from the work [11]. Consequently, we show how to construct $(2+1)$\-
and, in principle, higher-dimensional bi-Hamiltonian hierarchies on the
centrally extended Lie algebras associated to Novikov algebras. This
construction is formulated with respect to a Killing form. Finally, we
illustrate the above theory by explicit examples of $(2+1)$\- and
$(3+1)$-dimensional (dispersive) integrable systems and related bi-Hamiltonian
structures.

##  2 Novikov algebras

Let $(\mathbb{A},\cdot)$ be a Novikov algebra,222In this article all algebraic
structures are considered over the field of complex numbers $\mathbb{C}$. this
means that it is right-commutative

$\displaystyle({\bm{a}}\cdot{\bm{b}})\cdot{\bm{c}}=(\bm{a}\cdot{\bm{c}})\cdot{\bm{b}},$

and left-symmetric (quasi-associative)

$\displaystyle({\bm{a}}\cdot{\bm{b}})\cdot{\bm{c}}-{\bm{a}}\cdot({\bm{b}}\cdot{\bm{c}})=({\bm{b}}\cdot{\bm{a}})\cdot{\bm{c}}-{\bm{b}}\cdot({\bm{a}}\cdot{\bm{c}}).$

The structure constants of a Novikov algebra $\mathbb{A}$, with basis vectors
${\bm{e}}_{1},\ldots,{\bm{e}}_{n}$, are given by $b^{i}_{jk}$ such that

$\displaystyle{\bm{e}}_{i}\cdot{\bm{e}}_{j}=\sum_{k}b^{k}_{ij}{\bm{e}}_{k},\qquad\mathcal{B}\equiv\bigg{(}\sum_{k}b^{k}_{ij}{\bm{e}}_{k}\bigg{)},$

where $\mathcal{B}$ is the characteristic matrix.

By $\mathscr{L}_{\mathbb{A}}$ we understand the algebra of $\mathbb{A}$-valued
(smooth) functions of (independent) spatial variables $x,y_{1},\ldots,y_{N}$
belonging to the domain
$\Omega=\mathbb{S}^{1}\times\dots\times\mathbb{S}^{1}$,

$\displaystyle\mathscr{L}_{\mathbb{A}}:=\\{{\bm{u}}\colon\Omega\rightarrow\mathbb{A}\,|\,\text{${\bm{u}}$
is smooth}\\}.$

As the regular dual algebra $\mathscr{L}_{\mathbb{A}}^{*}$ to
$\mathscr{L}_{\mathbb{A}}$ we choose the vector space of
$\mathbb{A}^{*}$-valued functions,

$\displaystyle\mathscr{L}_{\mathbb{A}}^{*}:=\\{\bm{\xi}\colon\Omega\rightarrow\mathbb{A}^{*}\,|\,\text{$\bm{\xi}$
is smooth}\\}.$

The related duality pairing
$\mathscr{L}_{\mathbb{A}}^{*}\times\mathscr{L}_{\mathbb{A}}\rightarrow\mathbb{C}$
is given by

$\displaystyle\langle\bm{\xi},{\bm{u}}\rangle:=\int_{\Omega}(\bm{\xi},{\bm{u}}){\rm
d}\bm{x},$

where the integral is over all spatial variables, ${\rm d}\bm{x}\equiv{\rm
d}x{\rm d}y_{1}\cdots{\rm d}y_{N}$. By
$(\,,\,)\colon\mathbb{A}^{*}\times\mathbb{A}\rightarrow\mathbb{C}$ we mean the
natural pairing between $\mathbb{A}$ and its dual $\mathbb{A}^{*}$.

The Lie algebra structure on $\mathscr{L}_{\mathbb{A}}$ is defined, with
respect to the distinguished variable $x$, by the bracket

$\displaystyle[\\![{\bm{u}},{\bm{v}}]\\!]:={\bm{u}}_{x}\cdot{\bm{v}}-{\bm{v}}_{x}\cdot{\bm{u}},\qquad{\bm{u}}_{x}\equiv\frac{\partial{\bm{u}}}{\partial
x}.$ (2.1)

In fact, (2.1) is a Lie bracket on $\mathscr{L}_{\mathbb{A}}$ iff
$(\mathbb{A},\cdot)$ is a Novikov algebra.

##  3 Central extensions

Consider a bilinear form
$g\colon\mathbb{A}\times\mathbb{A}\rightarrow\mathbb{C}$, then we will say
that:

  * •

the form satisfies the quasi-Frobenius condition if

$\displaystyle
g({\bm{a}},{\bm{b}}\cdot{\bm{c}})=g({\bm{a}}\cdot{\bm{c}},{\bm{b}}),$

  * •

the form satisfies the cyclic condition if

$\displaystyle
g({\bm{a}},{\bm{b}}\cdot{\bm{c}})+g({\bm{b}},{\bm{c}}\cdot{\bm{a}})+g({\bm{c}},{\bm{a}}\cdot{\bm{b}})=0,$

  * •

the form is totally symmetric if the trilinear form

$\displaystyle
c({\bm{a}},{\bm{b}},{\bm{c}}):=g({\bm{a}}\cdot{\bm{b}},{\bm{c}})$

is symmetric with respect to all arguments.

We are interested in the central extensions of the Lie algebra
$\mathscr{L}_{\mathbb{A}}$. This means that on the direct sum
$\widehat{\mathscr{L}}_{\mathbb{A}}\equiv{\mathscr{L}_{\mathbb{A}}}\oplus\mathbb{C}$
there is a Lie bracket of the form

$\displaystyle[\\![{\bm{u}}\oplus\alpha,{\bm{v}}\oplus\beta]\\!]:=[\\![{\bm{u}},{\bm{v}}]\\!]\oplus\omega({\bm{u}},{\bm{v}}),$

where
$\omega\colon{\mathscr{L}_{\mathbb{A}}}\times{\mathscr{L}_{\mathbb{A}}}\rightarrow\mathbb{C}$
is a $2$-cocycle. This means that $\omega$ is skew-symmetric,

$\displaystyle\omega({\bm{u}},{\bm{v}})=-\omega({\bm{v}},{\bm{u}}),$

and it satisfies the cyclic condition

$\displaystyle\omega\bigl{(}[\\![{\bm{u}},{\bm{v}}]\\!],{\bm{w}}\bigr{)}+\omega\bigl{(}[\\![{\bm{v}},{\bm{w}}]\\!],{\bm{u}}\bigr{)}+\omega\bigl{(}[\\![{\bm{w}},{\bm{u}}]\\!],{\bm{v}}\bigr{)}=0.$

The differential $2$-cocycles, yielding central extensions of the Lie algebras
${\mathscr{L}_{\mathbb{A}}}$ associated with Novikov algebras, are generated
by appropriate bilinear forms satisfying various algebraic conditions that
were originally derived in [4], see also [21]:

  * •

A bilinear form $g\colon\mathbb{A}\times\mathbb{A}\rightarrow\mathbb{C}$
defines the first-order $2$-cocycle

$\displaystyle\omega({\bm{u}},{\bm{v}})=\int_{\Omega}g({\bm{u}}_{x},{\bm{v}}){\rm
d}\bm{x}$ (3.1a) iff $g$ is symmetric and satisfies the quasi-Frobenius
condition.

  * •

A bilinear form $f\colon\mathbb{A}\times\mathbb{A}\rightarrow\mathbb{C}$
defines the second-order $2$-cocycle

$\displaystyle\omega({\bm{u}},{\bm{v}})=\int_{\Omega}f({\bm{u}}_{xx},{\bm{v}}){\rm
d}\bm{x}$ (3.1b)

iff $f$ is skew-symmetric and satisfies the quasi-Frobenius and cyclic
conditions.

  * •

A bilinear form $h\colon\mathbb{A}\times\mathbb{A}\rightarrow\mathbb{C}$
defines the third-order $2$-cocycle

$\displaystyle\omega({\bm{u}},{\bm{v}})=\int_{\Omega}h({\bm{u}}_{xxx},{\bm{v}}){\rm
d}\bm{x}$ (3.1c)

iff $h$ is symmetric and the related trilinear form
$c({\bm{a}},{\bm{b}},{\bm{c}})\equiv h({\bm{a}}\cdot{\bm{b}},{\bm{c}})$ is
totally symmetric.

There are no $2$-cocycles of higher order.

Our aim is to complete the above theory with $2$-cocycles by which one can
introduce a new independent variable.

######  Theorem 3.1.

A bilinear form $\eta\colon\mathbb{A}\times\mathbb{A}\rightarrow\mathbb{C}$
generates on $\mathscr{L}_{\mathbb{A}}$ the first-order $2$-cocycle:

$\displaystyle\omega({\bm{u}},{\bm{v}}):=\int_{\Omega}\eta({\bm{u}}_{y},{\bm{v}}){\rm
d}\bm{x},$ (3.2)

defined with respect to an $($additional$)$ independent variable
$y\in\mathbb{S}^{1}$, if and only if the form $\eta$ is symmetric, satisfies
the quasi-Frobenius and cyclic conditions.333These conditions can be directly
obtained from the algebraic conditions for the multi-dimensional Poisson
brackets of Dubrovin–Novikov type obtained in [14, 15], see also [20].

###### Proof.

Integrating by parts

$\displaystyle\omega({\bm{u}},{\bm{v}}):=-\int_{\Omega}\eta({\bm{u}},{\bm{v}}_{y}){\rm
d}\bm{x}.$

Then,

$\displaystyle\omega\bigl{(}[\\![{\bm{u}},{\bm{v}}]\\!],{\bm{w}}\bigr{)}+\omega\bigl{(}[\\![{\bm{v}},{\bm{w}}]\\!],{\bm{u}}\bigr{)}+\omega\bigl{(}[\\![{\bm{w}},{\bm{u}}]\\!],{\bm{v}}\bigr{)}$
$\displaystyle\quad{}=-\int_{\Omega}\big{[}\eta({\bm{u}}_{x}\cdot{\bm{v}}-{\bm{v}}_{x}\cdot{\bm{u}},{\bm{w}}_{y})+\eta({\bm{v}}_{x}\cdot{\bm{w}}-{\bm{w}}_{x}\cdot{\bm{v}},{\bm{u}}_{y})+\eta({\bm{w}}_{x}\cdot{\bm{u}}-{\bm{u}}_{x}\cdot{\bm{w}},{\bm{v}}_{y})\big{]}{\rm
d}\bm{x}$
$\displaystyle\quad{}=\int_{\Omega}\big{[}\eta({\bm{u}}_{xy}\cdot{\bm{v}},{\bm{w}})+\eta({\bm{u}}_{x}\cdot{\bm{v}}_{y},{\bm{w}})-\eta({\bm{v}}_{xy}\cdot{\bm{u}},{\bm{w}})-\eta({\bm{v}}_{x}\cdot{\bm{u}}_{y},{\bm{w}})-\eta({\bm{v}}_{x}\cdot{\bm{w}},{\bm{u}}_{y})$
$\displaystyle\quad\quad{}\\!-\eta({\bm{w}}\cdot{\bm{v}}_{x},{\bm{u}}_{y})-\eta({\bm{w}}\cdot{\bm{v}},{\bm{u}}_{xy})+\eta({\bm{w}}\cdot{\bm{u}}_{x},{\bm{v}}_{y})+\eta({\bm{u}}\cdot{\bm{u}},{\bm{v}}_{xy})+\eta({\bm{u}}_{x}\cdot{\bm{w}},{\bm{v}}_{y})\big{]}{\rm
d}\bm{x}=0.$

Collecting terms with respect to the functionally independent variables, for
instance $\\{{\bm{u}}_{xy},{\bm{v}},{\bm{w}}\\}$, we obtain the required
conditions on the bilinear form $\eta$. ∎

Notice that always a linear combination of $2$-cocyles is a $2$-cocycle. This
means, that each linearly independent $2$-cocycle of the form (3.2) can be
defined with respect to a different additional independent variable.

In the article [2] the three- and less-dimensional Novikov algebras over
complex numbers were fully classified. In arbitrary dimension the
classification is far from being complete. For instance, in dimension four
only transitive Novikov algebras were classified [3, 7]. From the point of
view of the construction of integrable hierarchies from Novikov algebras the
transitive444A Novikov algebra is called transitive (or right-nilpotent) if
every right multiplication is nilpotent. algebras are not of interest as they
result in ’degenerate’ systems of evolution equations, see [21].

Recently, in the work [11] a classification of non-degenerate two-dimensional
Hamiltonian operators of Dubrovin–Novikov type (1.2) ($D=2$) with small number
of components, $n\leqslant 4$, extending previous results from [14, 15]
($n=1,2$), has been completed. Since all $2$-dimensional Hamiltonian operators
from that class can be reduced to a linear form the results of [11] provides
also an implicit classification of associated Novikov algebras for which there
exists first-order central extensions (3.1a) and (3.2) defined by non-
degenerate bilinear forms $g$ and $\eta$. Taking advantage of the
classification presented in [11] we have extracted in Table 2 all the
respective unique $4$-dimensional non-transitive Novikov algebras.

For a given Novikov algebra classification of the bilinear forms generating
differential central extensions is rather straightforward as it involves only
solving the systems of linear equations, that can be done without much
difficulty using any software for symbolic computations. The classification of
differential $2$-cocycles, with respect to variable $x$, associated with the
low-dimensional Novikov algebras was presented in the work [21]. Here, we
extend this classification over the case including differential $2$-cocycles
of the form (3.2), defined with respect to an additional independent variable.
The results are summarized in the following theorem and Tables 1 and 2. We in
fact are only interested with the cases of (3.2) defined by means of non-
degenerate $\eta$.

######  Theorem 3.2.

We consider here only Novikov algebras $\mathbb{A}$ that cannot be decomposed
into direct sum of lower-dimensional Novikov algebras.

  * •

In dimension $1$ there is only one relevant Novikov algebra,
$\mathbb{A}=\mathbb{C}$, and in this case there is no central extension of the
form (3.2).

  * •

In dimension $2$ there is only one Novikov algebra, $(N6)$ with $\kappa=-2$,
for which there exists the central extension (3.2) defined by a non-degenerate
bilinear form $\eta$.

  * •

In dimension $3$ there are only two Novikov algebras, $(A7)$ and $(D6)$ both
with $\kappa=-2$, for which there exists the central extension (3.2) defined
by a non-degenerate $\eta$.

  * •

In dimension $4$ there are several Novikov algebras for which there exists the
central extension (3.2) defined by a non-degenerate $\eta$.

  * •

Up to dimension $4$ there is only one transitive Novikov algebra,
$(N3)\otimes(N6)$ with $\kappa=-2$, for which the dimension of the linear
space spanned by central extensions (3.2) is higher than $1$.

######  Remark 3.3.

In the work [11] it is showed that any two-dimensional irreducible nonconstant
Hamiltonian operator of Dubrovin–Novikov type (1.2) ($D=2$) with three-
components ($n=3$) can be reduced to one of two canonical forms. These forms
correspond respectively to Novikov algebras $(A7)$ and $(D6)$ both with
$\kappa=-2$. By another result from [11] any non-degenerated three-dimensional
irreducible Hamiltonian operator (1.2) $(D=3)$ with three-components, which is
not transformable to constant coefficients, can be brought to one canonical
form. This form corresponds to the transitive algebra $(A7)$ with $\kappa=-2$.

##  4 Killing form

The Killing form on ${\mathscr{L}_{\mathbb{A}}}$ is a symmetric non-degenerate
bilinear form
$\mathcal{K}\colon{\mathscr{L}_{\mathbb{A}}}\times{\mathscr{L}_{\mathbb{A}}}\rightarrow\mathbb{C}$
which is $ad$-invariant:

$\displaystyle\mathcal{K}\bigl{(}[\\![{\bm{u}},{\bm{w}}]\\!],{\bm{v}}\bigr{)}+\mathcal{K}\bigl{(}{\bm{w}},[\\![{\bm{u}},{\bm{v}}]\\!]\bigr{)}=0.$

Assume that $\mathcal{K}$ is defined by a bilinear form
$\kappa\colon\mathbb{A}\times\mathbb{A}\rightarrow\mathbb{C}$:

$\displaystyle\mathcal{K}\bigl{(}{\bm{u}},{\bm{v}}\bigr{)}:=\int_{\Omega}\kappa({\bm{u}},{\bm{v}}){\rm
d}\bm{x}\equiv\langle\hat{\kappa}{\bm{u}},{\bm{v}}\rangle,$ (4.1)

where $\hat{\kappa}\colon\mathbb{A}\rightarrow\mathbb{A}^{*}$. If there exists
a Killing form then the dual Lie algebra ${\mathscr{L}_{\mathbb{A}}}^{*}$ can
be identified with ${\mathscr{L}_{\mathbb{A}}}$ and the co-adjoint action can
be identified with the adjoint action:

$\displaystyle\operatorname{ad}^{*}_{\bm{u}}\bm{\xi}\equiv\hat{\kappa}[\\![{\bm{u}},{\bm{w}}]\\!],\qquad\bm{\xi}\equiv\hat{\kappa}{\bm{w}},\qquad\langle\operatorname{ad}_{\bm{u}}^{*}\bm{\xi},{\bm{v}}\rangle:=-\langle\bm{\xi},[\\![{\bm{u}},{\bm{v}}]\\!]\rangle,$

where $\bm{\xi}\in{\mathscr{L}_{\mathbb{A}}}^{*}$.

######  Theorem 4.1.

A symmetric non-degenerate bilinear form
$\kappa\colon\mathbb{A}\times\mathbb{A}\rightarrow\mathbb{C}$ generates the
Killing form (4.1) if and only if the form $\kappa$ satisfies the cyclic and
quasi-Frobenius conditions.

###### Proof.

We have

$\displaystyle\mathcal{K}\bigl{(}[\\![{\bm{u}},{\bm{w}}]\\!],{\bm{u}}\bigr{)}+\mathcal{K}\bigl{(}{\bm{w}},[\\![{\bm{u}},{\bm{v}}]\\!]\bigr{)}$
$\displaystyle\quad{}=\int_{\Omega}\big{[}\kappa({\bm{u}}_{x}\cdot{\bm{w}}-{\bm{w}}_{x}\cdot{\bm{u}},{\bm{v}})+\kappa({\bm{w}},{\bm{u}}_{x}\cdot{\bm{v}}-{\bm{v}}_{x}\cdot{\bm{u}})\big{]}{\rm
d}\bm{x}$
$\displaystyle\quad{}=\int_{\Omega}\big{[}\kappa({\bm{u}}_{x}\cdot{\bm{w}},{\bm{v}})+\kappa({\bm{w}}\cdot{\bm{u}}_{x},{\bm{v}})+\kappa({\bm{w}}\cdot{\bm{u}},{\bm{v}}_{x})+\kappa({\bm{w}},{\bm{u}}_{x}\cdot{\bm{v}})-\kappa({\bm{w}},{\bm{v}}_{x}\cdot{\bm{u}})\big{]}{\rm
d}\bm{x}=0.$

Collecting terms with respect to functionally independent variables we obtain
two conditions

$\displaystyle\kappa({\bm{a}}\cdot{\bm{c}},{\bm{b}})+\kappa({\bm{c}}\cdot{\bm{a}},{\bm{b}})+\kappa({\bm{c}},{\bm{a}}\cdot{\bm{b}})=0$

and

$\displaystyle\kappa({\bm{c}}\cdot{\bm{a}},{\bm{b}})=\kappa({\bm{c}},{\bm{b}}\cdot{\bm{a}}).$

Taking into account the symmetry of the form $\kappa$ we get the conditions
from the theorem. ∎

We see that the algebraic conditions for the Killing forms (4.1) are the same
as for the $2$-cocycles (3.2) with non-degenerate $\eta$. This means that on a
Novikov algebra, on which there exists a ‘non-degenerate’ $2$-cocycle (3.2),
one can always define the Killing form (4.1).

In the following lemma we rewrite the algebraic conditions that must be
satisfied by differential central extensions with respect to a given Killing
form.

######  Lemma 4.2.

We assume that the Novikov algebra $\mathbb{A}$ is such that there on
${\mathscr{L}_{\mathbb{A}}}$ exists the associated Killing form (4.1). Then

  * •

a linear form $\hat{g}\colon\mathbb{A}\rightarrow\mathbb{A}$ generates the
two-cocycle of first order,

$\displaystyle\omega({\bm{u}},{\bm{v}})=\int_{\Omega}g({\bm{u}}_{x},{\bm{v}}){\rm
d}\bm{x}\equiv\mathcal{K}\big{(}\hat{g}{\bm{u}}_{x},{\bm{v}}\big{)},$

iff the following $($equivalent$)$ relations hold on $\mathbb{A}$

$\displaystyle\hat{g}({\bm{a}}\cdot{\bm{b}})=(\hat{g}{\bm{a}})\cdot{\bm{b}}$

and

$\displaystyle(\hat{g}{\bm{a}})\cdot{\bm{b}}-{\bm{a}}\cdot(\hat{g}{\bm{b}})=(\hat{g}{\bm{b}})\cdot{\bm{a}}-{\bm{b}}\cdot(\hat{g}{\bm{a}});$

  * •

a linear form $\hat{f}\colon\mathbb{A}\rightarrow\mathbb{A}$ generates the
two-cocycle of second order,

$\displaystyle\omega({\bm{u}},{\bm{v}})=\int_{\Omega}f({\bm{u}}_{xx},{\bm{v}}){\rm
d}\bm{x}\equiv\mathcal{K}\big{(}\hat{f}{\bm{u}}_{xx},{\bm{v}}\big{)},$

iff on $\mathbb{A}$ the following two conditions are satisfied,

$\displaystyle\hat{f}({\bm{a}}\cdot{\bm{b}})=\big{(}\hat{f}{\bm{a}}\big{)}\cdot{\bm{b}}$

and

$\displaystyle\hat{f}({\bm{a}}\cdot{\bm{b}}-{\bm{b}}\cdot{\bm{a}})={\bm{a}}\cdot\big{(}\hat{f}{\bm{b}}\big{)}+\big{(}\hat{f}{\bm{b}}\big{)}\cdot{\bm{a}};$

  * •

a linear form $\hat{h}\colon\mathbb{A}\rightarrow\mathbb{A}$ generates the
two-cocycle of third order,

$\displaystyle\omega({\bm{u}},{\bm{v}})=\int_{\Omega}h({\bm{u}}_{xxx},{\bm{v}}){\rm
d}\bm{x}\equiv\mathcal{K}\bigl{(}\hat{h}{\bm{u}}_{xxx},{\bm{v}}\bigr{)},$

iff the relations

$\displaystyle\hat{h}({\bm{a}}\cdot{\bm{b}})=\hat{h}({\bm{b}}\cdot{\bm{a}})=\big{(}\hat{h}{\bm{a}}\big{)}\cdot{\bm{b}}=\big{(}\hat{h}{\bm{b}}\big{)}\cdot{\bm{a}}$

and

$\displaystyle\hat{h}({\bm{a}}\cdot{\bm{b}})=-\frac{1}{2}{\bm{a}}\cdot\big{(}\hat{h}{\bm{b}}\big{)}$

are valid on $\mathbb{A}$;

  * •

a linear form $\hat{\eta}\colon\mathbb{A}\rightarrow\mathbb{A}$ generates the
two-cocycle of first order with respect to an additional independent variable
$y\in\mathbb{S}^{1}$,

$\displaystyle\omega({\bm{u}},{\bm{v}})=\int_{\Omega}\eta({\bm{u}}_{y},{\bm{v}}){\rm
d}\bm{x}\equiv\mathcal{K}\big{(}\hat{\eta}{\bm{u}}_{y},{\bm{v}}\big{)},$

iff

$\displaystyle\hat{\eta}({\bm{a}}\cdot{\bm{b}})=\big{(}\hat{\eta}{\bm{a}}\big{)}\cdot{\bm{b}}$

and

$\displaystyle(\hat{\eta}{\bm{a}})\cdot{\bm{b}}={\bm{a}}\cdot\big{(}\hat{\eta}{\bm{b}}\big{)}$

hold on $\mathbb{A}$.

We skip the proof as it is rather straightforward and it does not add anything
significant for further considerations.

##  5 Bi-Hamiltonian structure

Consider the centrally extended Lie–Poisson bracket, associated with the Lie
algebra ${\mathscr{L}_{\mathbb{A}}}$, and defined with respect to the Killing
form (4.1):

$\displaystyle\left\\{\mathcal{H},\mathcal{F}\right\\}_{1}({\bm{u}})=\mathcal{K}\bigl{(}{\bm{u}},[\\![\delta_{\bm{u}}\mathcal{H},\delta_{\bm{u}}\mathcal{F}]\\!]\bigr{)}+\omega_{1}\bigl{(}\delta_{\bm{u}}\mathcal{H},\delta_{\bm{u}}\mathcal{F}\bigr{)},$
(5.1)

where ${\bm{u}}\in{\mathscr{L}_{\mathbb{A}}}$ and
$\mathcal{H},\mathcal{F}\in\mathscr{F}\bigl{(}{\mathscr{L}_{\mathbb{A}}}\bigr{)}$.
Here, $\mathscr{F}\bigl{(}{\mathscr{L}_{\mathbb{A}}}\bigr{)}$ is the space of
functionals on the Lie algebra ${\mathscr{L}_{\mathbb{A}}}$:

$\displaystyle\mathcal{H}({\bm{u}})=\int_{\Omega}H[{\bm{u}}]{\rm d}\bm{x},$

where the densities $H[{\bm{u}}]$ are (non-local) smooth functions with
respect to all variables555Here, $\partial_{x}^{-1}$ is a formal inverse of
$\partial_{x}$ and its definition depends on the analytic assumptions which
are not significant from the point of view of the algebraic formalism.

$\displaystyle[{\bm{u}}]\equiv\big{\\{}{\bm{u}},{\bm{u}}_{x},{\bm{u}}_{y},{\bm{u}}_{xx},{\bm{u}}_{xy},\ldots,\partial_{x}^{-1}{\bm{u}}_{y},\ldots\big{\\}}.$

The respective (variational) differentials $\delta_{\bm{u}}\mathcal{H}$ are
defined through the standard formula

$\displaystyle\mathcal{K}\bigl{(}\delta_{\bm{u}}\mathcal{H},{\bm{v}}\bigr{)}:=\frac{{\rm
d}}{{\rm
d}\varepsilon}\mathcal{H}({\bm{u}}+\varepsilon{\bm{v}})\Bigr{|}_{\varepsilon=0},\qquad{\bm{u}},{\bm{v}}\in{\mathscr{L}_{\mathbb{A}}}.$
(5.2)

The second Poisson bracket compatible with (5.1) is

$\displaystyle\left\\{\mathcal{H},\mathcal{F}\right\\}_{0}({\bm{u}})=\omega_{0}\bigl{(}\delta_{\bm{u}}\mathcal{H},\delta_{\bm{u}}\mathcal{F}\bigr{)}.$
(5.3)

In the formulas (5.1) and (5.3) $\omega_{1}$ and $\omega_{0}$ are linear
compositions of admissible $2$-cocyles. Naturally, the Poisson brackets (5.1)
and (5.3) are compatible since linear composition of $\omega_{1}$ and
$\omega_{0}$ is also a $2$-cocycle.

The related Poisson tensors
$\mathcal{P}_{i}\colon{\mathscr{L}_{\mathbb{A}}}\rightarrow{\mathscr{L}_{\mathbb{A}}}$,
defined with respect to the Killing form (4.1),

$\displaystyle\left\\{\mathcal{H},\mathcal{F}\right\\}_{i}({\bm{u}})\equiv\mathcal{K}\bigl{(}\mathcal{P}_{i}\delta_{\bm{u}}\mathcal{H},\delta_{\bm{u}}\mathcal{F}\bigr{)},\qquad
i=1,2,$

are

$\displaystyle\mathcal{P}_{1}\bm{\gamma}=[\\![{\bm{u}},\bm{\gamma}]\\!]+\mathcal{I}_{1}\bm{\gamma}_{x}$
(5.4a) and
$\displaystyle\mathcal{P}_{0}\bm{\gamma}=\mathcal{I}_{0}\bm{\gamma}_{x}.$
(5.4b)

The linear operators
$\mathcal{I}_{i}\colon{\mathscr{L}_{\mathbb{A}}}\rightarrow{\mathscr{L}_{\mathbb{A}}}$,
such that

$\displaystyle\mathcal{K}\bigl{(}\mathcal{I}_{i}\bm{\gamma}_{x},{\bm{v}}\bigr{)}:=\omega_{i}\bigl{(}\bm{\gamma},{\bm{v}}\bigr{)},\qquad
i=0,1,$

have the form

$\displaystyle\mathcal{I}_{0}=\hat{g}_{0}+\hat{f}_{0}\partial_{x}+\hat{h}_{0}\partial_{x}^{2}+\sum_{k=1}^{N}\hat{\eta}_{0,k}\partial_{x}^{-1}\partial_{y_{k}},$
(5.5a) and
$\displaystyle\mathcal{I}_{1}=\hat{g}_{1}+\hat{f}_{1}\partial_{x}+\hat{h}_{1}\partial_{x}^{2}+\sum_{k=1}^{N}\hat{\eta}_{1,k}\partial_{x}^{-1}\partial_{y_{k}}.$
(5.5b)

The forms $\hat{g}_{i}$, $\hat{f}_{i}$, $\hat{h}_{i}$ and $\hat{\eta}_{i,k}$
generate the respective $2$-cocycles, see Lemma 4.2, and $N$ is the dimension
of the linear space spanned by $2$-cocycles of type (3.2) introducing
additional spatial variables.

The related (infinite) bi-Hamiltonian chain has the form

$\displaystyle{\bm{u}}_{t_{0}}=\mathcal{P}_{0}\delta_{\bm{u}}\mathcal{H}_{0}\equiv
0,$
$\displaystyle{\bm{u}}_{t_{1}}=\mathcal{P}_{1}\delta_{\bm{u}}\mathcal{H}_{0}=\mathcal{P}_{0}\delta_{\bm{u}}\mathcal{H}_{1},$
$\displaystyle{\bm{u}}_{t_{2}}=\mathcal{P}_{1}\delta_{\bm{u}}\mathcal{H}_{1}=\mathcal{P}_{0}\delta_{\bm{u}}\mathcal{H}_{2},$
(5.6) $\displaystyle\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots,$

where $\mathcal{H}_{i}\in\mathscr{F}({\mathscr{L}_{\mathbb{A}}})$ and
$\mathcal{H}_{0}$ is a Casimir of $\mathcal{P}_{0}$. Such bi-Hamiltonian chain
exists since $\mathcal{P}_{0}$ is a constant Poisson operator, which always
has non-trivial kernel. The evolution equations from the hierarchy are of
$(N+2)$ dimension: with respect to an evolution variable $t_{i}$ and $N+1$
spatial variables $x,y_{1},\ldots,y_{N}$.

Assuming invertibility of
$\mathcal{I}_{0}\colon{\mathscr{L}_{\mathbb{A}}}\rightarrow{\mathscr{L}_{\mathbb{A}}}$
we can introduce the auxiliary dependent variable ${\bm{\bar{u}}}$ such that

$\displaystyle{\bm{\bar{u}}}:=\mathcal{I}_{0}^{-1}{\bm{u}}\quad\iff\quad{\bm{u}}=\mathcal{I}_{0}{\bm{\bar{u}}}.$
(5.7)

Here, we understand the invertibility of the operator $\mathcal{I}_{0}$ as of
the pseudo-differential operators.

######  Theorem 5.1.

We assume that the Novikov algebra $\mathbb{A}$ possesses the right unity
${\bm{e}}$. Assuming that $\delta_{\bm{u}}\mathcal{H}_{0}=\bm{e}$, the first
two nontrivial evolution equations from the hierarchy (5.6) takes the form

$\displaystyle{\bm{u}}_{t_{1}}={\bm{u}}_{x},$
$\displaystyle{\bm{u}}_{t_{2}}=[\\![{\bm{u}},{\bm{\bar{u}}}]\\!]+\mathcal{I}_{1}{\bm{\bar{u}}}_{x}.$
(5.8)

The respective Hamiltonians are

$\displaystyle\mathcal{H}_{0}=\mathcal{K}\bigl{(}\bm{e},{\bm{u}}\bigr{)},$
$\displaystyle\mathcal{H}_{1}=\frac{1}{2}\mathcal{K}\bigl{(}{\bm{u}},{\bm{\bar{u}}}\bigr{)},$
$\displaystyle\mathcal{H}_{2}=\frac{1}{3}\mathcal{K}\bigl{(}[\\![{\bm{\bar{u}}},{\bm{u}}]\\!],\partial_{x}^{-1}{\bm{\bar{u}}}\bigr{)}+\frac{1}{2}\mathcal{K}\bigl{(}\mathcal{I}_{1}{\bm{\bar{u}}},{\bm{\bar{u}}}\bigr{)}.$
(5.9)

###### Proof.

The kernel of $\mathcal{P}_{0}$ is spanned by $x$-independent elements of
${\mathscr{L}_{\mathbb{A}}}$. For simplicity, we choose that
$\delta_{\bm{u}}\mathcal{H}_{0}=\bm{e}$. Thus,

$\displaystyle{\bm{u}}_{t_{1}}=\mathcal{P}_{1}\bm{e}\equiv{\bm{u}}_{x}=\mathcal{P}_{0}\delta_{\bm{u}}\mathcal{H}_{1}\equiv\mathcal{I}_{0}(\delta_{\bm{u}}\mathcal{H}_{1})_{x},$

and solving for $\delta_{\bm{u}}\mathcal{H}_{1}$ we find that
$\delta_{\bm{u}}\mathcal{H}_{1}={\bm{\bar{u}}}$. Hence,

$\displaystyle{\bm{u}}_{t_{2}}=\mathcal{P}_{1}{\bm{\bar{u}}}\equiv[\\![{\bm{u}},{\bm{\bar{u}}}]\\!]+\mathcal{I}_{1}{\bm{\bar{u}}}_{x}=\mathcal{P}_{0}\delta_{\bm{u}}\mathcal{H}_{2}\equiv\mathcal{I}_{0}(\delta_{\bm{u}}\mathcal{H}_{2})_{x}$

and we have666Applying (5.4a) to $\delta_{\bm{u}}\mathcal{H}_{2}$ one can
obtain the third flow,
${\bm{u}}_{t_{3}}=\mathcal{P}_{1}\delta_{\bm{u}}\mathcal{H}_{2}$, from the
hierarchy (5.6), but it does not have such a ‘neat’ form as previous flows.

$\displaystyle\delta_{\bm{u}}\mathcal{H}_{2}=\partial_{x}^{-1}\mathcal{I}_{0}^{-1}[\\![{\bm{u}},{\bm{\bar{u}}}]\\!]+\mathcal{I}_{0}^{-1}\mathcal{I}_{1}{\bm{\bar{u}}}.$

The Hamiltonians $\mathcal{H}_{i}$ related to the cosymmetries
$\delta_{\bm{u}}\mathcal{H}_{i}$ can be constructed using the homotopy
formula, see [5, 16] and proof of Theorem 1 in [21]. Alternatively, one can
verify the Hamiltonians (5.9) using the formula (5.2), but this way is less
straightforward as one needs to use the particular properties of the
respective $2$-cocycles. ∎

The evolution equations (5.8) and the Hamiltonians (5.9) can be written, in a
more explicit form, using the auxiliary variable ${\bm{\bar{u}}}$, see
Appendix A.

######  Remark 5.2.

Observe that $\mathcal{H}_{1}$ from (5.9) is a quadratic Hamiltonian
functional:

$\displaystyle\mathcal{H}_{1}=\frac{1}{2}\mathcal{K}\bigl{(}\mathcal{I}_{0}{\bm{\bar{u}}},{\bm{\bar{u}}}\bigr{)}\equiv\frac{1}{2}\mathcal{K}\big{(}{\bm{u}},\mathcal{I}_{0}^{-1}{\bm{u}}\big{)}.$

The operator $\mathcal{I}_{0}$ (5.5a), when it is invertible, can be
interpreted as the inertia operator. The variables ${\bm{u}}$ and
${\bm{\bar{u}}}$ play the role of the momentum and the velocity. As result,
the second non-trivial flow in the hierarchy (5.9) can be interpreted as the
integrable Euler equation on the centrally extended Lie algebra
${\mathscr{L}_{\mathbb{A}}}$ corresponding to the geodesic flow on the space
with metric defined by means of the inertia operator $\mathcal{I}_{0}$.
Compare this with Remark 1 and Appendix A.3 in the article [21]. For more
information on the subject we refer the reader to the books [1, 13].

##  6 Examples

###  6.1 Algebra $\bm{(N6)}$ for $\bm{\kappa=-2}$

This Novikov algebra is $2$-dimensional, non-abelian and non-associative, it
is also non-transitive. The associated Lie algebra
${\mathscr{L}_{\mathbb{A}}}$ with the Lie bracket (2.1) is isomorphic to the
Lie algebra
$\operatorname{Vect}\big{(}\mathbb{S}^{1}\big{)}\ltimes\operatorname{Vect}^{*}\big{(}\mathbb{S}^{1}\big{)}$,
that is the semidirect sum of the algebra of smooth vector fields on the
circle $\mathbb{S}^{1}$, $\operatorname{Vect}\big{(}\mathbb{S}^{1}\big{)}$,
with its dual $\operatorname{Vect}^{*}\big{(}\mathbb{S}^{1}\big{)}$, see [17]
and [18].

The structure matrix and the multiplication of the algebra $\mathbb{A}$ are

$\displaystyle\mathcal{B}=\begin{pmatrix}{\bm{e}}_{1}&-2{\bm{e}}_{2}\\\
{\bm{e}}_{2}&0\end{pmatrix},\qquad\begin{pmatrix}a_{1}\\\
a_{2}\end{pmatrix}\cdot\begin{pmatrix}b_{1}\\\
b_{2}\end{pmatrix}=\begin{pmatrix}a_{1}b_{1}\\\
a_{2}b_{1}-2a_{1}b_{2}\end{pmatrix}.$

The right unity is ${\bm{e}}=(1,0)^{\rm T}$. Taking the advantage of the
classification of the form $\eta$ in Table 1, we can define the Killing form
(4.1) by

$\displaystyle\hat{\kappa}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}.$

For ${\bm{u}}\in{\mathscr{L}_{\mathbb{A}}}$ and the related differential of a
functional $\mathcal{H}$ we choose

$\displaystyle{\bm{u}}=\begin{pmatrix}u(x,y)\\\
v(x,y)\end{pmatrix},\qquad\delta_{{\bm{u}}}\mathcal{H}=\begin{pmatrix}\dfrac{\delta
H}{\delta v}\vspace{1mm}\\\ \dfrac{\delta H}{\delta u}\end{pmatrix},$

as then we get the usual Euclidean duality:

$\displaystyle\mathcal{K}\bigl{(}\delta_{{\bm{u}}}{\mathcal{H}},{\bm{u}}_{t}\bigr{)}=\int_{\Omega}\left(\frac{\delta
H}{\delta u}u_{t}+\frac{\delta H}{\delta v}v_{t}\right){\rm d}\bm{x},$

where $\Omega=\mathbb{S}^{1}\times\mathbb{S}^{1}$ and ${\rm d}\bm{x}\equiv{\rm
d}x{\rm d}y$.

Using the classification of central extensions in Table 1 the operators (5.5)
are

$\displaystyle\mathcal{I}_{0}=\begin{pmatrix}g_{1}&0\\\
g_{2}&g_{1}\end{pmatrix}+\begin{pmatrix}0&0\\\
h&0\end{pmatrix}\partial_{x}^{2}+\begin{pmatrix}\eta&0\\\
0&\eta\end{pmatrix}\partial_{x}^{-1}\partial_{y}$

and

$\displaystyle\mathcal{I}_{1}=\begin{pmatrix}0&0\\\
\beta&0\end{pmatrix}\partial_{x}^{2}+\begin{pmatrix}\gamma&0\\\
0&\gamma\end{pmatrix}\partial_{x}^{-1}\partial_{y}.$

In this case the first-order cocycle is trivial, this means that in the
Poisson tensor (5.4a) $\hat{g}_{1}$ can be obtained/removed through a shift of
the dependent variables. So, we do not consider $\hat{g}_{1}$ in
$\mathcal{I}_{1}$. The relation (5.7) between the momentum variable ${\bm{u}}$
and the velocity variable ${\bm{\bar{u}}}=(\bar{u}(x,y),\bar{v}(x,y))^{\rm T}$
is

$\displaystyle u=g_{1}\bar{u}+\eta\partial_{x}^{-1}\bar{u}_{y},$
$\displaystyle
v=g_{1}\bar{v}+\eta\partial_{x}^{-1}\bar{v}_{y}+g_{2}\bar{u}+h\bar{u}_{xx}.$

The second flow from the hierarchy (5.6) takes the form

$\displaystyle u_{t_{2}}=u_{x}\bar{u}-u\bar{u}_{x}+\gamma\bar{u}_{y},$
$\displaystyle
v_{t_{2}}=\bar{u}v_{x}-u\bar{v}_{x}-2u_{x}\bar{v}+2\bar{u}_{x}v+\gamma\bar{v}_{y}+\beta\bar{u}_{xxx}.$

The related Poisson tensors (5.4) are

$\displaystyle\mathcal{P}_{0}=\begin{pmatrix}0&g_{1}\partial_{x}+\eta\partial_{y}\\\
g_{1}\partial_{x}+\eta\partial_{y}&g_{2}\partial_{x}+h\partial_{x}^{3}\end{pmatrix}$

and

$\displaystyle\mathcal{P}_{1}=\begin{pmatrix}0&u_{x}-u\partial_{x}+\alpha_{1}\partial_{x}+\gamma\partial_{y}\\\
-2u_{x}-u\partial_{x}+\alpha_{1}\partial_{x}+\gamma\partial_{y}&v_{x}+2v\partial_{x}+\alpha_{2}\partial_{x}+\beta\partial_{x}^{3}\end{pmatrix}.$

Notice that the above Poisson tensors are defined with respect to the standard
duality pairing. The Hamiltonian functionals (5.9) are

$\displaystyle\mathcal{H}_{0}=\int_{\Omega}v{\rm d}\bm{x},$
$\displaystyle\mathcal{H}_{1}=\frac{1}{2}\int_{\Omega}\bigl{(}\bar{u}v+u\bar{v}\bigr{)}{\rm
d}\bm{x},$
$\displaystyle\mathcal{H}_{2}=\int_{\Omega}\left[\frac{1}{3}\bigl{(}2u_{x}\bar{v}-\bar{u}v_{x}-2\bar{u}_{x}v+u\bar{v}_{x}\bigr{)}\partial_{x}^{-1}\bar{u}+\frac{1}{3}\bigl{(}u\bar{u}_{x}-u_{x}\bar{u}\bigr{)}\partial_{x}^{-1}\bar{v}\right.$
$\displaystyle\left.\hphantom{\mathcal{H}_{2}=}{}+\frac{1}{2}\big{(}\gamma\bar{u}\partial_{x}^{-1}\bar{v}_{y}+\beta\bar{u}\bar{u}_{xx}+\gamma\bar{v}\partial_{x}^{-1}\bar{u}_{y}\big{)}\right]{\rm
d}\bm{x}.$

The particular cases $\eta_{1}=\eta_{2}=h=0$ and $g_{1}=g_{2}=\beta=0$ of the
above bi-Hamiltonian structure were considered in the work [18], see the two
examples therein.

###  6.2 Algebra $\bm{(D6)}$ for $\bm{\kappa=-2}$

This Novikov algebra can be considered as the $3$-dimensional extension of the
algebra from the previous section as the subalgebra spanned by
$\\{{\bm{e}}_{1},{\bm{e}}_{2}\\}$ is identical to the algebra $(N6)$
($\kappa=-2$).

We proceed as before. The structure matrix and multiplication are

$\displaystyle\mathcal{B}=\begin{pmatrix}{\bm{e}}_{1}&-2{\bm{e}}_{2}&-\frac{1}{2}{\bm{e}}_{3}\\\
{\bm{e}}_{2}&0&0\\\
{\bm{e}}_{3}&0&-\frac{1}{2}{\bm{e}}_{2}\end{pmatrix},\qquad\begin{pmatrix}a_{1}\\\
a_{2}\\\ a_{3}\end{pmatrix}\cdot\begin{pmatrix}b_{1}\\\ b_{2}\\\
b_{3}\end{pmatrix}=\begin{pmatrix}a_{1}b_{1}\\\
a_{2}b_{1}-2a_{1}b_{2}-\frac{1}{2}a_{3}b_{3}\\\
a_{3}b_{1}-\frac{1}{2}a_{1}b_{3}\end{pmatrix}.$

The right unity is ${\bm{e}}=(1,0,0)^{\rm T}$. The Killing form (4.1) is
defined by

$\displaystyle\hat{\kappa}:=\begin{pmatrix}0&1&0\\\ 1&0&0\\\
0&0&1\end{pmatrix}.$

For ${\bm{u}}\in{\mathscr{L}_{\mathbb{A}}}$ and related differentials we
choose

$\displaystyle{\bm{u}}=\begin{pmatrix}u(x,y)\\\ v(x,y)\\\
w(x,y)\end{pmatrix},\qquad\delta_{{\bm{u}}}{\mathcal{H}}=\begin{pmatrix}\dfrac{\delta
H}{\delta v}\vspace{2mm}\\\ \dfrac{\delta H}{\delta u}\vspace{2mm}\\\
\dfrac{\delta H}{\delta w}\end{pmatrix},$

as then we have

$\displaystyle\mathcal{K}\bigl{(}\delta_{{\bm{u}}}{\mathcal{H}},{\bm{u}}_{t}\bigr{)}=\int_{\Omega}\left(\frac{\delta
H}{\delta u}u_{t}+\frac{\delta H}{\delta v}v_{t}+\frac{\delta H}{\delta
w}w_{t}\right){\rm d}\bm{x},$

where $\Omega=\mathbb{S}^{1}\times\mathbb{S}^{1}$ and ${\rm d}\bm{x}\equiv{\rm
d}x{\rm d}y$. The operators (5.5) are

$\displaystyle\mathcal{I}_{0}=\begin{pmatrix}g_{1}&0&0\\\ g_{2}&g_{1}&g_{3}\\\
g_{3}&0&g_{1}\end{pmatrix}+\begin{pmatrix}0&0&0\\\ h&0&0\\\
0&0&0\end{pmatrix}\partial_{x}^{2}+\begin{pmatrix}\eta&0&0\\\ 0&\eta&0\\\
0&0&\eta\end{pmatrix}\partial_{x}^{-1}\partial_{y}$

and

$\displaystyle\mathcal{I}_{1}=\begin{pmatrix}0&0&0\\\ \beta&0&0\\\
0&0&0\end{pmatrix}\partial_{x}^{2}+\begin{pmatrix}\gamma&0&0\\\ 0&\gamma&0\\\
0&0&\gamma\end{pmatrix}\partial_{x}^{-1}\partial_{y}.$

The relation (5.7) for the auxiliary variables
${\bm{\bar{u}}}=(\bar{u}(x,y),\bar{v}(x,y),\bar{w}(x,y))^{\rm T}$ is

$\displaystyle u=g_{1}\bar{u}+\eta\partial_{x}^{-1}\bar{u}_{y},$
$\displaystyle
v=g_{2}\bar{u}+g_{1}\bar{v}+g_{3}\bar{w}+\eta\partial_{x}^{-1}\bar{v}_{y}+h\bar{u}_{xx},$
$\displaystyle w=g_{3}\bar{u}+g_{1}\bar{w}+\eta\partial_{x}^{-1}\bar{w}_{y}.$

Then, we find the Poisson tensors (5.4)

$\displaystyle\mathcal{P}_{0}=\begin{pmatrix}0&g_{1}\partial_{x}+\eta\partial_{y}&0\\\
g_{1}\partial_{x}+\eta\partial_{y}&g_{2}\partial_{x}+h\partial_{x}^{3}&g_{3}\partial_{x}\\\
0&g_{3}\partial_{x}&g_{1}\partial_{x}+\eta\partial_{y}\end{pmatrix}$

and

$\displaystyle\mathcal{P}_{1}=\begin{pmatrix}0&u_{x}-u\partial_{x}+\gamma\partial_{y}&0\\\
-2u_{x}-u\partial_{x}+\gamma\partial_{y}&v_{x}+2v\partial_{x}+\beta\partial_{x}^{3}&-\frac{1}{2}w_{x}+\frac{1}{2}w\partial_{x}\\\
0&w_{x}+\frac{1}{2}w\partial_{x}&-\frac{1}{2}u_{x}-u\partial_{x}+\gamma\partial_{y}\end{pmatrix},$

the second system from the hierarchy (5.8)

$\displaystyle u_{t_{2}}=u_{x}\bar{u}-u\bar{u}_{x}+\gamma\bar{u}_{y},$
$\displaystyle
v_{t_{2}}=-2u_{x}\bar{v}+\bar{u}v_{x}+2u_{x}v-u\bar{v}_{x}-\frac{1}{2}w_{x}\bar{w}+\frac{1}{2}w\bar{w}_{x}+\gamma\bar{v}_{y}+\beta\bar{u}_{xxx},$
$\displaystyle
w_{t_{2}}=-\frac{1}{2}u_{x}\bar{w}+\bar{u}w_{x}+\frac{1}{2}\bar{u}_{x}w-u\bar{w}_{x}+\gamma\bar{w}_{y},$

and the related Hamiltonians (5.9)

$\displaystyle\mathcal{H}_{0}=\int_{\Omega}v{\rm d}\bm{x},$
$\displaystyle\mathcal{H}_{1}=\frac{1}{2}\int_{\Omega}\bigl{(}\bar{u}v+u\bar{v}+w\bar{w}\bigr{)}{\rm
d}\bm{x},$
$\displaystyle\mathcal{H}_{2}=\int_{\Omega}\left[\frac{1}{3}\left(2u_{x}\bar{v}-\bar{u}v_{x}-2\bar{u}_{x}v+u\bar{v}_{x}+\frac{1}{2}w_{x}\bar{w}-\frac{1}{2}w\bar{w}_{x}\right)\partial_{x}^{-1}\bar{u}\right.$
$\displaystyle\hphantom{\mathcal{H}_{2}=}{}+\frac{1}{3}\bigl{(}u\bar{u}_{x}-u_{x}\bar{u}\bigr{)}\partial_{x}^{-1}\bar{v}+\frac{1}{3}\left(\frac{1}{2}u_{x}\bar{w}-\bar{u}w_{x}-\frac{1}{2}\bar{u}_{x}w+u\bar{w}_{x}\right)\partial_{x}^{-1}\bar{w}$
$\displaystyle\left.\hphantom{\mathcal{H}_{2}=}{}+\frac{1}{2}\big{(}\gamma\bar{u}\partial_{x}^{-1}\bar{v}_{y}+\beta\bar{u}\bar{u}_{xx}+\gamma\bar{v}\partial_{x}^{-1}\bar{u}_{y}+\gamma\bar{w}\partial_{x}^{-1}\bar{w}_{y}\big{)}\right]{\rm
d}\bm{x}.$

Taking $w=\bar{w}=0$ and $g_{3}=0$ the above evolution equations and the
related bi-Hamiltonian structure reduce to the case from the previous section.

###  6.3 Algebra $\bm{(N3)\otimes(N6)}$ for $\bm{\kappa=-2}$

This Novikov algebra is the tensor product of the associative and abelian
algebra $(N3)$ with the Novikov algebra $(N6)$ for $\kappa=-2$. It is non-
abelian, non-associative, and also non-transitive. Up to dimension $4$ this is
the only non-transitive Novikov algebra that allows for the construction of
the $(3+1)$-dimensional bi-Hamiltonian hierarchy (5.6).

The structure matrix and algebra multiplication are:

$\displaystyle\mathcal{B}=\begin{pmatrix}{\bm{e}}_{1}&-2{\bm{e}}_{2}&{\bm{e}}_{3}&-2{\bm{e}}_{4}\\\
{\bm{e}}_{2}&0&{\bm{e}}_{4}&0\\\ {\bm{e}}_{3}&-2{\bm{e}}_{4}&0&0\\\
{\bm{e}}_{4}&0&0&0\end{pmatrix},\qquad\begin{pmatrix}a_{1}\\\ a_{2}\\\
a_{3}\\\ a_{4}\end{pmatrix}\cdot\begin{pmatrix}b_{1}\\\ b_{2}\\\ b_{3}\\\
b_{4}\end{pmatrix}=\begin{pmatrix}a_{1}b_{1}\\\ a_{2}b_{1}-2a_{1}b_{2}\\\
a_{3}b_{1}+a_{1}b_{3}\\\
a_{4}b_{1}-2a_{3}b_{2}+a_{2}b_{3}-2a_{1}b_{4}\end{pmatrix}.$

The right unity is ${\bm{e}}=(1,0,0,0)^{\rm T}$.

As the Killing form (4.1) we can take

$\displaystyle\hat{\kappa}:=\begin{pmatrix}0&0&0&1\\\ 0&0&1&0\\\ 0&1&0&0\\\
1&0&0&0\end{pmatrix},$

thus ${\bm{u}}=\bigl{(}u(x,y,z),v(x,y,z),w(x,y,z),s(x,y,z)\bigr{)}^{\rm T}$
and $\delta_{{\bm{u}}}{\mathcal{H}}=\big{(}\frac{\delta H}{\delta
s},\frac{\delta H}{\delta w},\frac{\delta H}{\delta v},\frac{\delta H}{\delta
u}\big{)}^{\rm T}$ so that

$\displaystyle\mathcal{K}\bigl{(}\delta_{{\bm{u}}}{\mathcal{H}},{\bm{u}}_{t}\bigr{)}=\int_{\Omega}\left(\frac{\delta
H}{\delta u}u_{t}+\frac{\delta H}{\delta v}v_{t}+\frac{\delta H}{\delta
w}w_{t}+\frac{\delta H}{\delta s}s_{t}\right){\rm d}\bm{x},$

where $\Omega=\mathbb{S}^{1}\times\mathbb{S}^{1}\times\mathbb{S}^{1}$ and
${\rm d}\bm{x}\equiv{\rm d}x{\rm d}y{\rm d}z$.

In this case there are two linearly independent central extensions of the form
(3.2), see Table 2. Hence, we introduce two additional independent variables
$y$ and $z$. To simplify this example, we further consider only the central
extension of the type (3.1c) and (3.2).

As result, the operators (5.5) are

$\displaystyle\mathcal{I}_{0}=\begin{pmatrix}0&0&0&0\\\ h_{1}&0&0&0\\\
0&0&0&0\\\
h_{2}&0&h_{1}&0\end{pmatrix}\partial_{x}^{2}+\begin{pmatrix}\eta_{1}&0&0&0\\\
0&\eta_{1}&0&0\\\ 0&0&\eta_{1}&0\\\
0&0&0&\eta_{1}\end{pmatrix}\partial_{x}^{-1}\partial_{y}+\begin{pmatrix}0&0&0&0\\\
0&0&0&0\\\ \eta_{2}&0&0&0\\\
0&\eta_{2}&0&0\end{pmatrix}\partial_{x}^{-1}\partial_{z}$

and

$\displaystyle\mathcal{I}_{1}=\begin{pmatrix}0&0&0&0\\\ \beta_{1}&0&0&0\\\
0&0&0&0\\\
\beta_{2}&0&\beta_{1}&0\end{pmatrix}\partial_{x}^{2}+\begin{pmatrix}\gamma_{1}&0&0&0\\\
0&\gamma_{1}&0&0\\\ 0&0&\gamma_{1}&0\\\
0&0&0&\gamma_{1}\end{pmatrix}\partial_{x}^{-1}\partial_{y}+\begin{pmatrix}0&0&0&0\\\
0&0&0&0\\\ \gamma_{2}&0&0&0\\\
0&\gamma_{2}&0&0\end{pmatrix}\partial_{x}^{-1}\partial_{z}.$

Next, for
${\bm{\bar{u}}}=(\bar{u}(x,y),\bar{v}(x,y),\bar{w}(x,y),\bar{s}(x,y,z))^{\rm
T}$ the relation (5.7) is

$\displaystyle u=\eta_{1}\partial_{x}^{-1}\bar{u}_{y},$ $\displaystyle
v=\eta_{1}\partial_{x}^{-1}\bar{v}_{y}+h_{1}\bar{u}_{xx},$ $\displaystyle
w=\eta_{1}\partial_{x}^{-1}\bar{w}_{y}+\eta_{2}\partial_{x}^{-1}\bar{u}_{z},$
$\displaystyle
s=\eta_{1}\partial_{x}^{-1}\bar{s}_{y}+\eta_{2}\partial_{x}^{-1}\bar{v}_{z}+h_{1}\bar{w}_{xx}+h_{2}\bar{u}_{xx}.$

In this case, the three-dimensional compatible Poisson tensors (5.4) are

$\displaystyle\mathcal{P}_{0}=\begin{pmatrix}0&0&0&\eta_{1}\partial_{y}\\\
0&0&\eta_{1}\partial_{y}&h_{1}\partial_{x}^{3}\\\
0&\eta_{1}\partial_{y}&0&\eta_{2}\partial_{z}\\\
\eta_{1}\partial_{y}&h_{1}\partial_{x}^{3}&\eta_{2}\partial_{z}&h_{2}\partial_{x}^{3}\end{pmatrix}$

and

$\displaystyle\mathcal{P}_{1}=\begin{pmatrix}0&0&0&u_{x}-u\partial_{x}+\gamma_{1}\partial_{y}\\\
0&0&-2u_{x}-u\partial_{x}+\gamma_{1}\partial_{y}&v_{x}+2v\partial_{x}+\beta_{1}\partial_{x}^{3}\\\
0&u_{x}-u\partial_{x}+\gamma_{1}\partial_{y}&0&w_{x}-w\partial_{x}+\gamma_{1}\partial_{z}\\\
-2u_{x}-u\partial_{x}+\gamma_{2}\partial_{y}\\!&v_{x}+2v\partial_{x}+\beta_{1}\partial_{x}^{3}&-2w_{x}-w\partial_{x}+\gamma_{2}\partial_{z}&s_{x}+2s\partial_{x}+\beta_{2}\partial_{x}^{3}\end{pmatrix}.$

The second nontrivial system from the hierarchy (5.8) is

$\displaystyle u_{t_{2}}=u_{x}\bar{u}-u\bar{u}_{x}+\gamma_{1}\bar{u}_{y},$
$\displaystyle
v_{t_{2}}=-2u_{x}\bar{v}+\bar{u}v_{x}+2\bar{u}_{x}v-u\bar{v}_{x}+\gamma_{1}\bar{v}_{y}+\beta_{1}\bar{u}_{xxx},$
$\displaystyle
w_{t_{2}}=u_{x}\bar{w}+\bar{u}w_{x}-\bar{u}_{x}w-u\bar{w}_{x}+\gamma_{1}\bar{w}_{y}+\gamma_{2}\bar{u}_{z},$
$\displaystyle
s_{t_{2}}=\bar{u}s_{x}-2u_{x}\bar{s}+v_{x}\bar{w}-2\bar{v}w_{x}-u\bar{s}_{x}+2\bar{u}_{x}s-\bar{v}_{x}w+2v\bar{w}_{x}$
$\displaystyle\hphantom{s_{t_{2}}=}{}+\gamma_{1}\bar{s}_{y}+\gamma_{2}\bar{v}_{z}+\beta_{1}\bar{w}_{xxx}+\beta_{2}\bar{u}_{xxx}.$

Notice that this is a $(3+1)$-dimensional (dispersive) integrable bi-
Hamiltonian evolution system and it can be explicitly written either by means
of the components of the momentum variable ${\bm{u}}$ or the velocity variable
${\bm{\bar{u}}}$. The respective Hamiltonians (5.9) are

$\displaystyle\mathcal{H}_{0}=\int_{\Omega}s{\rm d}\bm{x},$
$\displaystyle\mathcal{H}_{1}=\frac{1}{2}\int_{\Omega}\bigl{(}u\bar{s}+\bar{u}s+\bar{v}w+v\bar{w}\bigr{)}{\rm
d}\bm{x},$

and

$\displaystyle\mathcal{H}_{2}=\int_{\Omega}\left[\frac{1}{3}\bigl{(}-\bar{u}s_{x}+2u_{x}\bar{s}+u\bar{s}_{x}-2\bar{u}_{x}s-v_{x}\bar{w}+2\bar{v}w_{x}+\bar{v}_{x}w-2v\bar{w}_{x}\bigr{)}\partial_{x}^{-1}\bar{u}\right.$
$\displaystyle\hphantom{\mathcal{H}_{2}=}{}-\frac{1}{3}\bigl{(}u_{x}\bar{w}+\bar{u}w_{x}-\bar{u}_{x}w-u\bar{w}_{x}\bigr{)}\partial_{x}^{-1}\bar{v}+\frac{1}{3}\bigl{(}2u_{x}\bar{v}-\bar{u}v_{x}-2\bar{u}_{x}v+u\bar{v}_{x}\bigr{)}\partial_{x}^{-1}\bar{w}$
$\displaystyle\hphantom{\mathcal{H}_{2}=}{}+\frac{1}{3}\bigl{(}u\bar{u}_{x}-u_{x}\bar{u}\bigr{)}\partial_{x}^{-1}\bar{s}+\frac{1}{2}\big{(}\gamma_{1}\bar{s}\partial_{x}^{-1}\bar{u}_{y}+\gamma_{1}\bar{w}\partial_{x}^{-1}\bar{v}_{y}+\gamma_{1}\bar{v}\partial_{x}^{-1}\bar{w}_{y}+\gamma_{2}\bar{v}\partial_{x}^{-1}\bar{u}_{z}\big{)}$
$\displaystyle\left.\hphantom{\mathcal{H}_{2}=}{}+\frac{1}{2}\big{(}\gamma_{1}\bar{u}\partial_{x}^{-1}\bar{s}_{y}+\gamma_{2}\bar{u}\partial_{x}^{-1}\bar{v}_{z}+\beta_{1}\bar{u}_{xx}\bar{w}+\beta_{1}\bar{u}\bar{w}_{xx}+\beta_{2}\bar{u}\bar{u}_{xx}\big{)}\right]{\rm
d}\bm{x}.$

Notice that in the example from this section one can reduce dimension taking
$y=z$ or $y=x$ or $z=x$.

##  Appendix A Explicit form of the Bi-Hamiltonian hierarchy

In the variable ${\bm{\bar{u}}}$ the bi-Hamiltonian chain (5.6) takes the form

$\displaystyle{\bm{\bar{u}}}_{t_{0}}=\bar{\mathcal{P}}_{0}\delta_{\bm{\bar{u}}}\mathcal{H}_{0}\equiv
0,$
$\displaystyle{\bm{\bar{u}}}_{t_{1}}=\bar{\mathcal{P}}_{1}\delta_{\bm{\bar{u}}}\mathcal{H}_{0}=\bar{\mathcal{P}}_{0}\delta_{\bm{\bar{u}}}\mathcal{H}_{1},$
$\displaystyle{\bm{\bar{u}}}_{t_{2}}=\bar{\mathcal{P}}_{1}\delta_{\bm{\bar{u}}}\mathcal{H}_{1}=\bar{\mathcal{P}}_{0}\delta_{\bm{\bar{u}}}\mathcal{H}_{2},$
$\displaystyle\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots,$

where
$\bar{\mathcal{P}}_{i}=\mathcal{I}_{0}^{-1}\mathcal{P}_{i}\mathcal{I}_{0}^{-1}$
and
$\delta_{\bm{\bar{u}}}\mathcal{H}=\mathcal{I}_{0}\delta_{\bm{u}}\mathcal{H}$.
Then,

$\displaystyle{\bm{\bar{u}}}_{t_{1}}={\bm{\bar{u}}}_{x},\qquad\mathcal{I}_{0}{\bm{\bar{u}}}_{t_{2}}=[\\![\mathcal{I}_{0}{\bm{\bar{u}}},{\bm{\bar{u}}}]\\!]+\mathcal{I}_{1}{\bm{\bar{u}}}_{x}$
(A.1)

and

$\displaystyle\mathcal{H}_{0}=\mathcal{K}\bigl{(}\bm{c},\mathcal{I}_{0}{\bm{\bar{u}}}\bigr{)},\qquad\mathcal{H}_{1}=\frac{1}{2}\mathcal{K}\bigl{(}\mathcal{I}_{0}{\bm{\bar{u}}},{\bm{\bar{u}}}\bigr{)},\qquad\mathcal{H}_{2}=\frac{1}{3}\mathcal{K}\big{(}[\\![{\bm{\bar{u}}},\mathcal{I}_{0}{\bm{\bar{u}}}]\\!],\partial_{x}^{-1}{\bm{\bar{u}}}\big{)}+\frac{1}{2}\mathcal{K}\bigl{(}\mathcal{I}_{1}{\bm{\bar{u}}},{\bm{\bar{u}}}\bigr{)}.$

######  Proposition A.1.

The explicit form, in the variable ${\bm{\bar{u}}}$, of the second flow from
(A.1) is

$\displaystyle\hat{g}_{0}{\bm{\bar{u}}}_{t_{2}}+\hat{f}_{0}{\bm{\bar{u}}}_{xt_{2}}+\hat{h}_{0}{\bm{\bar{u}}}_{xxt_{2}}+\sum_{k=1}^{N}\hat{\eta}_{0,k}\partial_{x}^{-1}{\bm{\bar{u}}}_{y_{k}t_{2}}$
$\displaystyle=\left(\frac{1}{2}\hat{g}_{0}{\bm{\bar{u}}}\cdot{\bm{\bar{u}}}-\frac{1}{2}{\bm{\bar{u}}}\cdot\hat{g}_{0}{\bm{\bar{u}}}+\hat{f}_{0}{\bm{\bar{u}}}_{x}\cdot{\bm{\bar{u}}}+\hat{h}_{0}{\bm{\bar{u}}}_{xx}\cdot{\bm{\bar{u}}}+\frac{1}{2}\hat{h}_{0}{\bm{\bar{u}}}_{x}\cdot{\bm{\bar{u}}}_{x}-\sum_{k=1}^{N}\hat{\eta}_{0,k}{\bm{\bar{u}}}\cdot\partial_{x}^{-1}{\bm{\bar{u}}}_{y_{k}}\right)_{x}$
$\displaystyle\quad{}+\sum_{k=1}^{N}\hat{\eta}_{0,k}({\bm{\bar{u}}}\cdot{\bm{\bar{u}}})_{y_{k}}+\hat{g}_{1}{\bm{\bar{u}}}_{x}+\hat{f}_{1}{\bm{\bar{u}}}_{xx}+\hat{h}_{1}{\bm{\bar{u}}}_{xxx}+\sum_{k=1}^{N}\hat{\eta}_{1,k}{\bm{\bar{u}}}_{y_{k}},$

and the explicit form of Hamiltonians is

$\displaystyle\mathcal{H}_{0}=\mathcal{K}\bigl{(}\hat{g}_{0}{\bm{c}},{\bm{\bar{u}}}\bigr{)},$
$\displaystyle\mathcal{H}_{1}=\frac{1}{2}\mathcal{K}\bigl{(}\hat{g}_{0}{\bm{\bar{u}}},{\bm{\bar{u}}}\bigr{)}+\frac{1}{2}\mathcal{K}\bigl{(}\hat{f}_{0}{\bm{\bar{u}}}_{x},{\bm{\bar{u}}}\bigr{)}+\frac{1}{2}\mathcal{K}\bigl{(}\hat{h}_{0}{\bm{\bar{u}}}_{xx},{\bm{\bar{u}}}\bigr{)}+\frac{1}{2}\sum_{k=1}^{N}\mathcal{K}\bigl{(}\hat{\eta}_{0,k}\partial_{x}^{-1}{\bm{\bar{u}}}_{y_{k}},{\bm{\bar{u}}}\bigr{)},$
$\displaystyle\mathcal{H}_{2}=\frac{1}{2}\mathcal{K}\bigl{(}\hat{g}_{0}{\bm{\bar{u}}},{\bm{\bar{u}}}\cdot{\bm{\bar{u}}}\bigr{)}+\frac{1}{3}\mathcal{K}\bigl{(}\hat{f}_{0}{\bm{\bar{u}}}_{x},{\bm{\bar{u}}}\cdot{\bm{\bar{u}}}\bigr{)}+\frac{1}{3}\mathcal{K}\bigl{(}\hat{h}_{0}{\bm{\bar{u}}}_{xx},{\bm{\bar{u}}}\cdot{\bm{\bar{u}}}\bigr{)}+\frac{1}{6}\mathcal{K}\bigl{(}\hat{h}_{0}{\bm{\bar{u}}}_{x},{\bm{\bar{u}}}\cdot{\bm{\bar{u}}}_{x}\bigr{)}$
$\displaystyle\hphantom{\mathcal{H}_{2}=}{}+\frac{1}{3}\sum_{k=1}^{N}\mathcal{K}\bigl{(}\hat{\eta}_{0,k}{\bm{\bar{u}}},\partial_{x}^{-1}{\bm{\bar{u}}}_{y_{k}}\cdot{\bm{\bar{u}}}\bigr{)}-\frac{1}{3}\sum_{k=1}^{N}\mathcal{K}\bigl{(}\hat{\eta}_{0,k}{\bm{\bar{u}}},{\bm{\bar{u}}}\cdot\partial_{x}^{-1}{\bm{\bar{u}}}_{y_{k}}\bigr{)}$
$\displaystyle\hphantom{\mathcal{H}_{2}=}{}+\frac{1}{2}\mathcal{K}\bigl{(}\hat{g}_{1}{\bm{\bar{u}}},{\bm{\bar{u}}}\bigr{)}+\frac{1}{2}\mathcal{K}\bigl{(}\hat{f}_{1}{\bm{\bar{u}}}_{x},{\bm{\bar{u}}}\bigr{)}+\frac{1}{2}\mathcal{K}\bigl{(}\hat{h}_{1}{\bm{\bar{u}}}_{xx},{\bm{\bar{u}}}\bigr{)}+\frac{1}{2}\sum_{k=1}^{N}\mathcal{K}\bigl{(}\hat{\eta}_{1,k}\partial_{x}^{-1}{\bm{\bar{u}}}_{y_{k}},{\bm{\bar{u}}}\bigr{)}.$

###### Proof.

We will skip the details. The proposition is a consequence of the following
relations

$\displaystyle[\\![\hat{g}_{0}{\bm{\bar{u}}},{\bm{\bar{u}}}]\\!]=(\hat{g}_{0}{\bm{\bar{u}}}_{x})\cdot{\bm{\bar{u}}}-{\bm{\bar{u}}}_{x}\cdot(\hat{g}_{0}{\bm{\bar{u}}})=\frac{1}{2}\bigl{(}\hat{g}_{0}{\bm{\bar{u}}}\cdot{\bm{\bar{u}}}-{\bm{\bar{u}}}\cdot\hat{g}_{0}{\bm{\bar{u}}}\bigr{)}_{x},$
$\displaystyle[\\![\hat{f}_{0}{\bm{\bar{u}}}_{x},{\bm{\bar{u}}}]\\!]=(\hat{f}_{0}{\bm{\bar{u}}}_{xx})\cdot{\bm{\bar{u}}}-{\bm{\bar{u}}}_{x}\cdot(\hat{f}_{0}{\bm{\bar{u}}}_{x})=\hat{f}_{0}({\bm{\bar{u}}}_{x}\cdot{\bm{\bar{u}}})_{x},$
$\displaystyle[\\![\hat{h}_{0}{\bm{\bar{u}}}_{xx},{\bm{\bar{u}}}]\\!]=(\hat{h}_{0}{\bm{\bar{u}}}_{xxx})\cdot{\bm{\bar{u}}}-{\bm{\bar{u}}}_{x}\cdot(\hat{h}_{0}{\bm{\bar{u}}}_{xx})=\hat{h}_{0}\left({\bm{\bar{u}}}_{xx}\cdot{\bm{\bar{u}}}+\frac{1}{2}{\bm{\bar{u}}}_{x}\cdot{\bm{\bar{u}}}_{x}\right)_{x},$
$\displaystyle[\\![\hat{\eta}_{0,k}\partial_{x}^{-1}{\bm{\bar{u}}}_{y_{k}},{\bm{\bar{u}}}]\\!]=(\hat{\eta}_{0,k}{\bm{\bar{u}}}_{y_{k}})\cdot{\bm{\bar{u}}}-{\bm{\bar{u}}}_{x}\cdot\big{(}\hat{\eta}_{0,k}\partial_{x}^{-1}{\bm{\bar{u}}}_{y_{k}}\big{)}=\hat{\eta}_{0,k}({\bm{\bar{u}}}\cdot{\bm{\bar{u}}})_{y_{k}}-\hat{\eta}_{0,k}\big{(}{\bm{\bar{u}}}\cdot\partial_{x}^{-1}{\bm{\bar{u}}}_{y_{k}}\big{)}_{x},$

which can be proven using all the properties of the linear forms generating
respective $2$-cocycles obtained in Lemma 4.2. ∎

The above proposition is the higher-dimensional extension of Theorem 1 from
the article [21], see also the equation (26) therein.

Table 1: The classification of the bilinear forms generating respective differential $2$-cocycles (3.1) and (3.2) for all non-decomposable $1$ and $2$-dimensional Novikov algebras and the significant $3$-dimensional cases corresponding to Theorem 3.2. We use for the Novikov algebras the same names as in [2]. type | $\mathcal{B}$ | $g$ | $f$ | $h$ | $\eta$ | comments
---|---|---|---|---|---|---
$\mathbb{C}$ | $e_{1}$ | $g_{11}$ | $0$ | $h_{11}$ | 0 |  $\begin{array}[]{@{}c@{}}\text{\small abelian}\\\\[-4.30554pt] \text{\small associative}\end{array}$2pt1pt
$(T2)$ | $\begin{pmatrix}{\bm{e}}_{2}&0\\\ 0&0\end{pmatrix}$ | $\begin{pmatrix}g_{11}&g_{12}\\\ g_{12}&0\end{pmatrix}$ | $\begin{pmatrix}0&0\\\ 0&0\end{pmatrix}$ | $\begin{pmatrix}h_{11}&h_{12}\\\ h_{12}&0\end{pmatrix}$ | $\begin{pmatrix}\eta_{11}&0\\\ 0&0\end{pmatrix}$ |  $\begin{array}[]{@{}c@{}}\text{\small abelian}\\\\[-4.30554pt] \text{\small associative}\\\\[-4.30554pt] \text{\small transitive}\end{array}$10pt
$(T3)$ | $\begin{pmatrix}0&0\\\ -{\bm{e}}_{1}&0\end{pmatrix}$ | $\begin{pmatrix}0&g_{12}\\\ g_{12}&g_{22}\end{pmatrix}$ | $\begin{pmatrix}0&0\\\ 0&0\end{pmatrix}$ | $\begin{pmatrix}0&0\\\ 0&h_{22}\end{pmatrix}$ | $\begin{pmatrix}0&0\\\ 0&\eta_{22}\end{pmatrix}$ |  transitive10pt
$(N3)$ | $\begin{pmatrix}{\bm{e}}_{1}&{\bm{e}}_{2}\\\ {\bm{e}}_{2}&0\end{pmatrix}$ | $\begin{pmatrix}g_{11}&g_{12}\\\ g_{12}&0\end{pmatrix}$ | $\begin{pmatrix}0&0\\\ 0&0\end{pmatrix}$ | $\begin{pmatrix}h_{11}&h_{12}\\\ h_{12}&0\end{pmatrix}$ | $\begin{pmatrix}0&0\\\ 0&0\end{pmatrix}$ |  $\begin{array}[]{@{}c@{}}\text{\small abelian}\\\\[-4.30554pt] \text{\small associative}\end{array}$10pt
$(N4)$ | $\begin{pmatrix}0&{\bm{e}}_{1}\\\ 0&{\bm{e}}_{2}\end{pmatrix}$ | $\begin{pmatrix}g_{11}&g_{12}\\\ g_{12}&g_{22}\end{pmatrix}$ | $\begin{pmatrix}0&f_{12}\\\ -f_{12}&0\end{pmatrix}$ | $\begin{pmatrix}0&0\\\ 0&h_{22}\end{pmatrix}$ | $\begin{pmatrix}0&0\\\ 0&0\end{pmatrix}$ |  associative10pt
$(N5)$ | $\begin{pmatrix}0&{\bm{e}}_{1}\\\ 0&{\bm{e}}_{1}+{\bm{e}}_{2}\\\ \end{pmatrix}$ | $\begin{pmatrix}0&g_{12}\\\ g_{12}&g_{22}\\\ \end{pmatrix}$ | $\begin{pmatrix}0&0\\\ 0&0\\\ \end{pmatrix}$ | $\begin{pmatrix}0&0\\\ 0&h_{22}\\\ \end{pmatrix}$ | $\begin{pmatrix}0&0\\\ 0&0\end{pmatrix}$ |  10pt
$\begin{array}[]{@{}c@{}}(N6)\\\\[-2.15277pt] \kappa\neq 0,1,-2\end{array}$ | $\begin{pmatrix}{\bm{e}}_{1}&\kappa{\bm{e}}_{2}\\\ {\bm{e}}_{2}&0\end{pmatrix}$ | $\begin{pmatrix}g_{11}&g_{12}\\\ g_{12}&0\end{pmatrix}$ | $\begin{pmatrix}0&0\\\ 0&0\\\ \end{pmatrix}$ | $\begin{pmatrix}h_{11}&0\\\ 0&0\\\ \end{pmatrix}$ | $\begin{pmatrix}0&0\\\ 0&0\end{pmatrix}$ |  10pt
$\begin{array}[]{@{}c@{}}(N6)\\\\[-2.15277pt] \kappa=-2\end{array}$ | $\begin{pmatrix}{\bm{e}}_{1}&-2{\bm{e}}_{2}\\\ {\bm{e}}_{2}&0\end{pmatrix}$ | $\begin{pmatrix}g_{11}&g_{12}\\\ g_{12}&0\end{pmatrix}$ | $\begin{pmatrix}0&0\\\ 0&0\end{pmatrix}$ | $\begin{pmatrix}h_{11}&0\\\ 0&0\end{pmatrix}$ | $\begin{pmatrix}0&\eta_{12}\\\ \eta_{12}&0\end{pmatrix}$ |  $\det\eta\neq 0$ 10pt
$\begin{array}[]{@{}c@{}}(A7)\\\\[-2.15277pt] \kappa\neq 1,-2\end{array}$ | $\begin{pmatrix}{\bm{e}}_{2}&\kappa{\bm{e}}_{3}&0\\\ {\bm{e}}_{3}&0&0\\\ 0&0&0\end{pmatrix}$ | $\begin{pmatrix}g_{11}&g_{12}&g_{13}\\\ g_{12}&g_{13}&0\\\ g_{13}&0&0\end{pmatrix}$ | $\begin{pmatrix}0&0&0\\\ 0&0&0\\\ 0&0&0\end{pmatrix}$ | $\begin{pmatrix}h_{11}&h_{12}&0\\\ h_{12}&0&0\\\ 0&0&0\end{pmatrix}$ | $\begin{pmatrix}\eta_{11}&0&0\\\ 0&0&0\\\ 0&0&0\end{pmatrix}$ |  transitive10pt
$\begin{array}[]{@{}c@{}}(A7)\\\\[-2.15277pt] \kappa=-2\end{array}$ | $\begin{pmatrix}{\bm{e}}_{2}&-2{\bm{e}}_{3}&0\\\ {\bm{e}}_{3}&0&0\\\ 0&0&0\end{pmatrix}$ | $\begin{pmatrix}g_{11}&g_{12}&g_{13}\\\ g_{12}&g_{13}&0\\\ g_{13}&0&0\end{pmatrix}$ | $\begin{pmatrix}0&0&0\\\ 0&0&0\\\ 0&0&0\end{pmatrix}$ | $\begin{pmatrix}h_{11}&h_{12}&0\\\ h_{12}&0&0\\\ 0&0&0\end{pmatrix}$ | $\begin{pmatrix}\eta_{11}&0&\eta_{13}\\\ 0&\eta_{13}&0\\\ \eta_{13}&0&0\end{pmatrix}$ |  $\begin{array}[]{@{}c@{}}\text{\small transitive}\\\\[-2.15277pt] \det\eta\neq 0\end{array}$10pt
$\begin{array}[]{@{}c@{}}(D6)\\\\[-2.15277pt] \kappa\neq 1,-2\end{array}$ | $\begin{pmatrix}{\bm{e}}_{1}&\kappa{\bm{e}}_{2}&\frac{\kappa+1}{2}{\bm{e}}_{3}\\\ {\bm{e}}_{2}&0&0\\\ {\bm{e}}_{3}&0&-\frac{1}{2}{\bm{e}}_{2}\end{pmatrix}$ | $\begin{pmatrix}g_{11}&-{\scriptstyle(\kappa+1)}g_{33}&g_{13}\\\ -{\scriptstyle(\kappa+1)}g_{33}&0&0\\\ g_{13}&0&g_{33}\end{pmatrix}$ | $\begin{pmatrix}0&0&{\scriptstyle\delta_{\kappa,-1}}f_{13}\\\ 0&0&0\\\ -{\scriptstyle\delta_{\kappa,-1}}f_{13}&0&0\end{pmatrix}$ | $\begin{pmatrix}h_{11}&0&0\\\ 0&0&0\\\ 0&0&0\end{pmatrix}$ | $\begin{pmatrix}0&0&{\scriptstyle\delta_{\kappa,-5}}\eta_{13}\\\ 0&0&0\\\ {\scriptstyle\delta_{\kappa,-5}}\eta_{13}&0&0\end{pmatrix}$ |  10pt
$\begin{array}[]{@{}c@{}}(D6)\\\\[-2.15277pt] \kappa=-2\end{array}$ | $\begin{pmatrix}{\bm{e}}_{1}&-2{\bm{e}}_{2}&-\frac{1}{2}{\bm{e}}_{3}\\\ {\bm{e}}_{2}&0&0\\\ {\bm{e}}_{3}&0&-\frac{1}{2}{\bm{e}}_{2}\end{pmatrix}$ | $\begin{pmatrix}g_{11}&g_{12}&g_{13}\\\ g_{12}&0&0\\\ g_{13}&0&g_{12}\end{pmatrix}$ | $\begin{pmatrix}0&0&0\\\ 0&0&0\\\ 0&0&0\end{pmatrix}$ | $\begin{pmatrix}h_{11}&0&0\\\ 0&0&0\\\ 0&0&0\end{pmatrix}$ | $\begin{pmatrix}0&\eta_{12}&0\\\ \eta_{12}&0&0\\\ 0&0&\eta_{12}\end{pmatrix}$ |  $\det\eta\neq 0$ 10pt
Table 2: The classification of the bilinear forms generating respective differential $2$-cocycles (3.1) and (3.2) for all non-decomposable non-transitive $4$-dimensional Novikov algebras with non-degenerate bilinear forms $g$ and $\eta$. The Novikov algebras are extracted from the work [11], where the corresponding Hamiltonian operators of hydrodynamic type are classified. All the following Novikov algebras are non-abelian and non-associative. type | $\mathcal{B}$ | $g$ | $f$ | $h$ | $\eta$
---|---|---|---|---|---
$\begin{array}[]{@{}c@{}}(N3)\otimes(N6)\\\\[-2.15277pt] \kappa=-2\end{array}$ | $\begin{pmatrix}{\bm{e}}_{1}&-2{\bm{e}}_{2}&{\bm{e}}_{3}&-2{\bm{e}}_{4}\\\ {\bm{e}}_{2}&0&{\bm{e}}_{4}&0\\\ {\bm{e}}_{3}&-2{\bm{e}}_{4}&0&0\\\ {\bm{e}}_{4}&0&0&0\end{pmatrix}$ | $\begin{pmatrix}g_{11}&g_{12}&g_{13}&g_{14}\\\ g_{12}&0&g_{14}&0\\\ g_{13}&g_{14}&0&0\\\ g_{14}&0&0&0\\\ \end{pmatrix}$ | $\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ \end{pmatrix}$ | $\begin{pmatrix}h_{11}&0&h_{13}&0\\\ 0&0&0&0\\\ h_{13}&0&0&0\\\ 0&0&0&0\end{pmatrix}$ |  $\begin{pmatrix}0&\eta_{12}&0&\eta_{14}\\\ \eta_{12}&0&\eta_{14}&0\\\ 0&\eta_{14}&0&0\\\ \eta_{14}&0&0&0\end{pmatrix}$ 10pt
| $\begin{pmatrix}{\bm{e}}_{1}&-2{\bm{e}}_{2}&0&-{\bm{e}}_{4}\\\ {\bm{e}}_{2}&0&0&0\\\ {\bm{e}}_{3}&0&0&-{\bm{e}}_{2}\\\ {\bm{e}}_{4}&0&0&0\end{pmatrix}$ | $\begin{pmatrix}g_{11}&g_{12}&g_{13}&g_{14}\\\ g_{12}&0&0&0\\\ g_{13}&0&g_{33}&g_{12}\\\ g_{14}&0&g_{12}&0\end{pmatrix}$ | $\begin{pmatrix}0&0&f_{13}&0\\\ 0&0&0&0\\\ -f_{13}&0&0&0\\\ 0&0&0&0\end{pmatrix}$ | $\begin{pmatrix}h_{11}&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{pmatrix}$ |  $\begin{pmatrix}0&\eta_{12}&0&0\\\ \eta_{12}&0&0&0\\\ 0&0&0&\eta_{12}\\\ 0&0&\eta_{12}&0\end{pmatrix}$ 10pt
| $\begin{pmatrix}{\bm{e}}_{1}&-2{\bm{e}}_{2}&0&-{\bm{e}}_{4}\\\ {\bm{e}}_{2}&0&0&0\\\ {\bm{e}}_{3}-{\bm{e}}_{4}&0&0&-{\bm{e}}_{2}\\\ {\bm{e}}_{4}&0&0&0\end{pmatrix}$ | $\begin{pmatrix}g_{11}&g_{12}&g_{13}&0\\\ g_{12}&0&0&0\\\ g_{13}&0&g_{33}&g_{12}\\\ 0&0&g_{12}&0\end{pmatrix}$ | $\begin{pmatrix}0&0&f_{13}&0\\\ 0&0&0&0\\\ -f_{13}&0&0&0\\\ 0&0&0&0\end{pmatrix}$ | $\begin{pmatrix}h_{11}&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{pmatrix}$ |  $\begin{pmatrix}0&\eta_{12}&0&0\\\ \eta_{12}&0&0&0\\\ 0&0&\eta_{12}&\eta_{12}\\\ 0&0&\eta_{12}&0\end{pmatrix}$ 10pt
| $\begin{pmatrix}{\bm{e}}_{1}+{\bm{e}}_{3}&-2{\bm{e}}_{2}&0&-2{\bm{e}}_{2}-{\bm{e}}_{4}\\\ {\bm{e}}_{2}&0&0&0\\\ {\bm{e}}_{3}&0&0&-{\bm{e}}_{2}\\\ {\bm{e}}_{2}+{\bm{e}}_{4}&0&0&0\end{pmatrix}$ | $\begin{pmatrix}g_{11}&g_{12}&g_{13}&g_{14}\\\ g_{12}&0&0&0\\\ g_{13}&0&0&g_{12}\\\ g_{14}&0&g_{12}&0\end{pmatrix}$ | $\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{pmatrix}$ | $\begin{pmatrix}h_{11}&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{pmatrix}$ |  $\begin{pmatrix}0&\eta_{12}&0&0\\\ \eta_{12}&0&0&0\\\ 0&0&0&\eta_{12}\\\ 0&0&\eta_{12}&0\end{pmatrix}$ 10pt
$\begin{array}[]{@{}c@{}}\kappa\geqslant-\frac{1}{2}\\\ \kappa\neq 0,1\end{array}$ | $\begin{pmatrix}{\bm{e}}_{1}&\kappa{\bm{e}}_{2}&-(\kappa+1){\bm{e}}_{3}&-2{\bm{e}}_{4}\\\ {\bm{e}}_{2}&0&-(\kappa+1){\bm{e}}_{4}&0\\\ {\bm{e}}_{3}&\kappa{\bm{e}}_{4}&0&0\\\ {\bm{e}}_{4}&0&0&0\end{pmatrix}$ | $\begin{pmatrix}g_{11}&g_{12}&g_{13}&g_{14}\\\ g_{12}&0&g_{14}&0\\\ g_{13}&g_{14}&0&0\\\ g_{14}&0&0&0\end{pmatrix}$ | $\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{pmatrix}$ | $\begin{pmatrix}h_{11}&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{pmatrix}$ |  $\begin{pmatrix}0&0&0&\eta_{14}\\\ 0&0&\eta_{14}&0\\\ 0&\eta_{14}&0&0\\\ \eta_{14}&0&0&0\end{pmatrix}$ 10pt
| $\begin{pmatrix}{\bm{e}}_{1}-2{\bm{e}}_{2}&{\bm{e}}_{3}&4{\bm{e}}_{4}-{\bm{e}}_{3}&-2{\bm{e}}_{4}\\\ {\bm{e}}_{2}-2{\bm{e}}_{3}&{\bm{e}}_{4}&-{\bm{e}}_{4}&0\\\ {\bm{e}}_{3}-2{\bm{e}}_{4}&0&0&0\\\ {\bm{e}}_{4}&0&0&0\end{pmatrix}$ | $\begin{pmatrix}g_{11}&g_{12}&g_{13}&g_{14}\\\ g_{12}&g_{13}&g_{14}&0\\\ g_{13}&g_{14}&0&0\\\ g_{14}&0&0&0\end{pmatrix}$ | $\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{pmatrix}$ | $\begin{pmatrix}h_{11}&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{pmatrix}$ |  $\begin{pmatrix}0&0&0&\eta_{14}\\\ 0&0&\eta_{14}&0\\\ 0&\eta_{14}&0&0\\\ \eta_{14}&0&0&0\end{pmatrix}$ 10pt

### Acknowledgements

I would like to thank Maciej Błaszak, Artur Sergyeyev and Ian Strachan for
various conversations concerning theory of higher-dimensional systems and/or
topics related to Novikov algebras. I would also like to thank the anonymous
referees for providing to my attention the reference [11] and some important
comments.

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