# KdV hierarchies and quantum Novikov equations.

V. M. Buchstaber$, A. V. Mikhailov♮ $ Steklov Mathematical Institute, RAS,
Moscow, Russia<EMAIL_ADDRESS>♮ University of Leeds, UK
<EMAIL_ADDRESS>
###### Abstract.

This paper begins with a review of the well-known KdV hierarchy, the $N$-th
Novikov equation, and its finite hierarchy in the classical commutative case.
This finite hierarchy consists of $N$ compatible integrable polynomial
dynamical systems in ${\mathbb{C}}^{2N}$. We discuss a non-commutative version
of the $N$-th Novikov hierarchy defined on the finitely generated free
associative algebra ${\mathfrak{B}}_{N}$ with $2N$ generators. Using the
method of quantisation ideals in ${\mathfrak{B}}_{N}$, for $N=1,2,3,4$, we
obtain two-sided homogeneous ideals
${\mathfrak{Q}}_{N}\subset{\mathfrak{B}}_{N}$ (quantisation ideals) that are
invariant with respect to the $N$-th Novikov equation and such that the
quotient algebra ${\mathfrak{C}}_{N}={\mathfrak{B}}_{N}/{\mathfrak{Q}}_{N}$
has a well-defined Poincare–Birkhoff–Witt basis. This allows us to define the
quantum $N$-th Novikov equation and its hierarchy on ${\mathfrak{C}}_{N}$. We
derive $N$ commuting quantum first integrals (Hamiltonians) and represent the
equations of the hierarchy in the Heisenberg form. Essential for our research
is the concept of cyclic Frobenius algebras, which we introduced in our recent
paper. In terms of the quadratic form that defines the structure of a cyclic
Frobenius algebra, we explicitly express the first integrals of the $N$-th
Novikov hierarchy in the commutative, free, and quantum cases.

###### Contents

  1. 1 Novikov equations and the corresponding finite KdV hierarchies.
    1. 1.1 Lie algebra of evolutionary differentiations.
    2. 1.2 Cyclic Frobenius algebras.
    3. 1.3 KdV hierarchy.
    4. 1.4 The $N$-th Novikov hierarchy.
  2. 2 KdV hierarchy and Novikov equations on free associative algebra.
    1. 2.1 KdV hierarchy on free associative algebra.
    2. 2.2 Cyclic Frobenius algebras over free associative algebra.
    3. 2.3 Non-commutative $N$-th Novikov equation and its hierarchy.
    4. 2.4 Self-adjointness of the KdV and $N$–Novikov hierarchies.
  3. 3 Quantisation of Novikov equations.
    1. 3.1 Quantum Novikov equations.
    2. 3.2 Quantum $N=1$ Novikov equation.
    3. 3.3 Quantum $N=2$ Novikov hierarchy.
    4. 3.4 Quantum $N=3$ and $N=4$ Novikov hierarchy.

## Introduction.

The problem of quantisation of dynamical systems has a history reaching back
over a century. In 1925 Heisenberg put forward a new quantum theory suggesting
that the multiplication rules, rather than the equations of classical
mechanics require modifications [18]. Almost immediately111The speed of
publications and Dirac’s reaction was astonishing! Heisenberg’s paper [18] was
submitted on 29th of July, published in September. Dirac received a proof of
Heisenberg’s paper in August, submitted his paper containing a deep
development of Heisenberg’s theory on 7th of November, which was published on
the 1-st of December 1925 [11]. Dirac reformulated Heisenberg’s ideas in
mathematical form, introduced quantum algebra, and noticed that in the limit
$\hbar\to 0$ the Heisenberg commutators of quantum observables tend to the
Poisson brackets between the corresponding observables in the classical
mechanics $\hat{a}\hat{b}-\hat{b}\hat{a}\to i\hbar\\{a,b\\}$ [11]. In other
words, non-commutative multiplication rules in the quantum theory can be
regarded as a deformation of commutative multiplication of smooth functions.
Nowadays there is enormous amount of papers, books and conferences devoted to
deformation quantisation. In [11] Dirac stated that “The correspondence
between the quantum and classical theories lies not so much in the limiting
agreement when $\hbar\to 0$ as in the fact that the mathematical operations on
the two theories obey in many cases the same laws” and raise the important
issue of self-consistency of the quantum multiplication rules and their
consistency with the equations of motion for finite value of the Plank
constant $\hbar$.

Recently, AVM presented a new approach to the problem of quantisation [21]. It
is proposed to start with dynamical systems defined on a free associative
algebra, i.e. with a free associative mechanics. In this theory smooth
functions on a phase space (or a Poisson manifold) are replaced by elements of
a free algebra, generated by the dynamical variables. Any finitely generated
associative algebra, including Dirac’s quantum algebra, is a quotient of a
free algebra with the same number of generators over a suitable two-sided
ideal. The commutation rules of a quantum theory enables one to swap positions
of any two variables. In [21] by quantisation it is understood a reduction of
a system defined on a free associative algebra to the dynamical system on a
quotient algebra such that any two generators can be re-ordered using its
multiplication rule. In order to achieve the consistency (to solve the issue
raised by Dirac) the ideal (the quantisation ideal) should be invariant with
respect to the derivation defined by the dynamical system. The classical
commutative case corresponds to the ideal generated by the commutators of all
dynamical variables. The method of quantisation proposed in [21] does not
appeal to a Poisson structure of the system, and therefore it enables to
define a concept of non-deformation quantisation. For example, the Volterra
integrable lattice admits a deformation quantisation. Using the new method it
is shown that its cubic symmetry admits two different quantisations, and one
of which is non-deformation. This approach has been developed further and
applied to quantisation of the Volterra hierarchy in [9]. In particular it was
shown that a periodic Volterra lattice with period three admits a bi-quantum
structure which is a quantum analog of the corresponding bi-Hamiltonian
structure.

The aim of our paper is to apply the approach proposed in [21] to the problem
of quantisation of stationary flows of the KdV hierarchy, known as the Novikov
equations [6], [10], [17], [24]. Novikov discovered that the stationary flows
of the KdV equation is a completely integrable dynamical system, it possess a
rich family of periodic and quasiperiodic exact solutions which can be
expressed in terms of Abelian functions [13],[24]. Here we would like to
emphasise that we study the problem of quantisation of finite dimensional
systems of ordinary differential equations and not of the field theory
associated with partial differential equations of the KdV hierarchy.

In Section 1 we give an explicit algebraic description of $N$-th Novikov
equation and the corresponding finite hierarchy of symmetries in the form
convenient for further generalisations. The $N$-the Novikov equation is an
ordinary differential equation of order $2N$. In Proposition 18 it is shown
that a complete set of $N$ first integrals of the $N$-th Novikov equations can
be explicitly presented in terms of the coefficients of fractional powers
$L^{\frac{2k-1}{2}}$ of the Schrödinger operator $L=D^{2}-u,\ D=\frac{d}{dx}$.
The KdV hierarchy defines evolutionary derivations in the graded algebra
${\mathfrak{A}}_{0}={\mathbb{C}}[u_{0},u_{1},u_{2},\ldots]$ with the weights
$|u_{k}|=k+2$, where $u_{0}=u,\,u_{k+1}=D(u_{k})$. The commuting evolutionary
derivations define a representation of algebraically independent variables
$u_{0},u_{1},u_{2},\ldots$ as smooth functions
$u_{k}=u_{k}(t_{1},t_{3},\ldots)$ of graded variables $t_{2k-1},\
k\in{\mathbb{N}}$ where the weight $|t_{2k-1}|=-2k+1$ and $t_{1}=x$. We treat
the $N$-th Novikov equation as a generator of a differential ideal
${\mathfrak{I}}_{N}$ in the graded ring
${\mathfrak{A}}={\mathcal{A}}[u,u_{1},\ldots]$, where ${\mathcal{A}}$ is a
commutative algebra of graded parameters
$\alpha_{4},\alpha_{6},\ldots,\alpha_{2N+2}$ where the weights
$|\alpha_{2n}|=2n$. The Proposition 17 shows that the KdV hierarchy induces
the finite $N$ hierarchy of integrable ordinary differential polynomial
equations on the quotient ring ${\mathfrak{A}}\diagup{\mathfrak{I}}_{N}$ which
is called $N$-th Novikov hierarchy. Here by integrability we understand the
existence of $N$ first integrals and $N$ commuting symmetries, one of which is
the $N$-th Novikov equation itself.

Let $\mathcal{B}$ be an associative $\mathbb{C}$-algebra with the unit 1, and
let $\mathcal{M}$ be some $\mathbb{C}$-linear space, $\dim\mathcal{M}\geqslant
1$. A cyclic Frobenius algebra ($\rm{CF}$-algebra) $\mathcal{B}$ is an algebra
$\mathcal{B}$ with a $\mathbb{C}$-bilinear skew-symmetric form
$\Phi(\cdot,\cdot)\colon\mathcal{B}\otimes_{\mathbb{C}}\mathcal{B}\to\mathcal{M}$
such that $\Phi(A,BC)+\Phi(B,CA)+\Phi(C,AB)=0$ where $A,B,C\in\mathcal{B}$,
see [7]. The choice of the name is motivated by the fact that the structure of
the $\rm{CF}$-algebra, as in the case of Frobenius algebras, is determined by
some quadratic form $\Phi$ and emphasizes a new property of the cyclicity of
the form $\Phi$ in our case. In partial cases when $\mathcal{M}=\mathbb{C}$,
our $\rm{CF}$-algebras are anti-Frobenius algebras. The latter was introduced
and developed in connection with the associative Yang-Baxter equation, see
[28]. In Section 1.2 we describe properties of the $\rm{CF}$-algebra in the
case $\mathcal{B}=\mathcal{M}={\mathfrak{A}}_{0}$. In terms of the form
$\Phi=\sigma(\cdot,\cdot)$ of this algebra, the first $N$-Novikov integrals
are explicitly described in the case of a commutative ring of polynomials
${\mathfrak{A}}_{0}$.

The KdV equations with non-commutative matrix variables were introduced in
[29], [8]. The KdV hierarchies on free associative algebras were studied in
[12], [14], [26], [27], [28]. In Section 2 we give a description of the
integrable KdV hierarchy on a differential graded free associative algebra
${\mathfrak{B}}_{0}={\mathbb{C}}\langle u,u_{1},\ldots\rangle,\
D(u_{k})=u_{k+1}$. Here, by integrability we understand the existence of an
infinite hierarchy of commuting symmetries, which are generators of symmetries
of the non-commutative KdV equation. There is a complete classification of
integrable hierarchies of evolutionary non-commutative equations [27]. In
particular, it was shown that the hierarchy of the KdV equation can be
generated by a (non-local) recursion operator. In the non-commutative case in
order to define local conservation laws we need to introduce a linear space of
functionals with the values in the quotient linear space
${\mathfrak{B}}_{0}\diagup\big{(}{\rm
Span}([{\mathfrak{B}}_{0},{\mathfrak{B}}_{0}])\oplus
D({\mathfrak{B}}_{0})\big{)}$, see [12], [26], [27]. Formal definitions of the
$N$-th Novikov equation and its hierarchy of symmetries are the same as in the
commutative case. Namely, we take a stationary flow of a linear combination of
the first $N$ members of the KdV hierarchy with commuting constant
coefficients $\alpha_{2n}\in{\mathcal{A}}$, as a generator of the two-sided
ideal ${\mathfrak{I}}_{N}\subset{\mathfrak{B}}={\mathcal{A}}\langle
u,u_{1}\ldots\rangle$. The $N$-Novikov hierarchy is defined as the canonical
projection of the KdV hierarchy to the quotient ring
${\mathfrak{B}}_{N}={\mathfrak{B}}\diagup{\mathfrak{I}}_{N}$ which is free
over ${\mathcal{A}}$ and finitely generated. The first system of the hierarchy
$\partial_{t_{1}}u_{k}={\mathcal{D}}(u_{k}),\ k=0,\ldots 2N-1$ is the $N$-th
Novikov equation itself, written in the form of a first order system where
${\mathcal{D}}$ is the derivation of ${\mathfrak{B}}_{N}$ induced by $D$. In
contrast to the commutative case, the hierarchy of linearly independent
symmetries is infinite. The case $N=1$ is already nontrivial. For $N=1$ the
Novikov equation coincides with the (non-commutative) Newton equation
$u_{2}=3u^{2}+8\alpha_{4}$ and in ${\mathfrak{B}}_{1}$ is represented by the
first order system

$\partial_{t_{1}}u=u_{1},\quad\partial_{t_{1}}u_{1}=3u^{2}+8\alpha_{4}.$ (1)

Equation (1) admits an infinite hierarchy of commuting symmetries. First four
of them are presented in Section 2.

A general definition of first integrals for equations on free associative
algebra was discussed in [22]. First integrals for the non-commutative $N$-th
Novikov equation and its hierarchy are introduced in Definition 25. In Section
2.2 we describe the properties of the $FH(\mathcal{B},\mathcal{M})$-algebra,
where $\mathcal{B}={\mathfrak{B}}_{0}$, and
$\mathcal{M}={\mathfrak{B}}_{0}/{\rm
Span}([{\mathfrak{B}}_{0},{\mathfrak{B}}_{0}])$. First integrals of the non-
commutative $N$-th Novikov hierarchy are explicitly represented in terms of
the form $\Phi=\sigma(\cdot,\cdot)$ of this algebra. Using (59), we
constructed infinitely many algebraically independent first integrals for the
non-commutative $N$-th Novikov equation and its hierarchy.

In Section 3 we consider the quantisation problem for $N$-th Novikov equation
following the method proposed in [21]. Let ${\mathcal{Q}}_{N}$ be a
commutative graded ring of parameters

${\mathcal{Q}}_{N}={\mathbb{C}}[\alpha_{2j+2},q_{i,j},q_{i,j}^{\omega}\,|\;0\leqslant
i<j\leqslant 2N-1,\;0\leqslant|\omega|<i+j+4]$

where $|q_{i,j}|=0$,
$\omega=(i_{2N-1},\ldots,i_{1},i_{0})\in{\mathbb{Z}}_{\geqslant}^{2N}$,
$|\omega|=(2N+1)i_{2N-1}+\cdots+3i_{1}+2i_{0}$,
$|q_{i,j}^{\omega}|=i+j+4-|\omega|$, and ${\mathfrak{B}}_{N}(q)$ denotes the
graded free associative ring ${\mathfrak{B}}_{N}(q)={\mathcal{Q}}_{N}\langle
u_{0},\ldots,u_{2N-1}\rangle$. Having $N$-th Novikov equation on
${\mathfrak{B}}_{N}(q)$, we introduce a differential homogeneous two-sided
ideal ${\mathfrak{Q}}_{N}\subset{\mathfrak{B}}_{N}(q)$ generated by the
polynomials

$p_{i,j}=u_{i}u_{j}-q_{i,j}u_{j}u_{i}+\sum\limits_{0\leqslant|\omega|<i+j+4}q_{i,j}^{\omega}u^{\omega},\quad
0\leqslant i<j\leqslant 2N-1,\;q_{i,j}\neq 0.$ (2)

where $u^{\omega}=u_{2N-1}^{i_{2N-1}}\cdots u_{1}^{i_{1}}u_{0}^{i_{0}}$ are
normally ordered monomials. The ideal ${\mathfrak{Q}}_{N}$ is a _quantisation
ideal_ of the $N$-th Novikov equation if the quotient algebra
${\mathfrak{C}}_{N}={\mathfrak{B}}_{N}(q)/{\mathfrak{Q}}_{N}$ has a
Poincaré–Birkhoff–Witt additive ${\mathcal{Q}}_{N}$-basis of normally ordered
monomials $u^{\omega},\ \omega\in{\mathbb{Z}}_{\geqslant}^{2N}$, and
${\mathfrak{Q}}_{N}$ is invariant with respect to the derivation
${\mathcal{D}}$. It follows from
${\mathcal{D}}({\mathfrak{Q}}_{N})\subset{\mathfrak{Q}}_{N}$ that the
coefficients $q_{i,j},q_{i,j}^{\omega}$ satisfy a system of algebraic
equations. In particular, these equation imply that $q_{i,j}=1$ for all
$0\leqslant i<j\leqslant 2N-1$ (Lemma 34). In the cases $N=1,2,3$ and $4$ we
have found out that the all structure constants $q_{i,j}^{\omega}$ of the
quantisation ideals ${\mathfrak{Q}}_{N}$ can be parameterised by one parameter
which we denote $\hbar$. In the case $N=1$ the computations are presented in
full detail in Section 3.2. In this case we have shown that the quantisation
ideal for equation (1) is generated by the commutation relation
$[u_{1},u]=i\hbar$, which coincides with Heisenberg’s commutation relation in
quantum mechanics [18], [11]. In the case $N=2$ we have shown (Proposition 41)
that the quantisation ideal ${\mathfrak{Q}}_{2}$ is generated by six
commutation relations

$\begin{array}[]{l}\phantom{.}[u_{i},u_{j}]=0\ \mbox{\rm for }\ i+j<3\
\mbox{\rm or }\ i+j=4;\\\
\phantom{.}[u_{3},u]=[u_{1},u_{2}]=i\hbar,\quad[u_{3},u_{2}]=10i\hbar
u_{0},\end{array}$

The quantum $N=2$ KdV hierarchy can be written in the Heisenberg form (Theorem
43)

$\displaystyle\partial_{t_{1}}u_{k}$
$\displaystyle=\;{\mathfrak{D}}(u_{k})=\frac{i}{\hbar}[{\mathfrak{H}}_{5,3},u_{k}]=\begin{cases}u_{k+1},\;0\leqslant
k\leqslant 2,\\\
32\alpha_{6}-16\alpha_{4}u+5u_{1}^{2}+10u_{2}u-10u^{3},\;k=3;\end{cases}$
$\displaystyle 4\partial_{t_{3}}u_{k}$
$\displaystyle=\;\frac{i}{\hbar}[{\mathfrak{H}}_{5,5},u_{k}]={\mathcal{D}}^{k+1}(u_{2}-3u^{2}).$

Here ${\mathfrak{H}}_{5,3},{\mathfrak{H}}_{5,5}\in{\mathfrak{C}}_{2}$ are
commuting quantum self-adjoint Hamiltonians (Lemma 80, Proposition 42). We
conclude Section 3 by discussion of quantum ideals for $N=3$ and $N=4$ and the
hierarchy of quantum KdV equations in the Heisenberg form in the case $N=3$.

We emphasize, that the method of quantisation proposed in [21] does not assume
any Hamiltonian structure of the noncomutative dynamical system, nevertheless
we present the quantum equations in the Heisenberg form
$\partial_{t}u_{k}=\frac{i}{\hbar}[{\mathfrak{H}},u_{k}]$ in Section 3.

##  1\. Novikov equations and the corresponding finite KdV hierarchies.

###  1.1. Lie algebra of evolutionary differentiations.

Consider a graded commutative differential polynomial algebra

${\mathfrak{A}}_{0}=(\mathbb{C}[u_{0},u_{1},\ldots],\ D),$ (3)

where $D$ is a derivation of $\mathbb{C}[u_{0},u_{1},\ldots]$ such that
$D(u_{k})=u_{k+1},\ k=0,1,\ldots$ In terms of grading we assume that the
variables $u_{k}$ have weight $|u_{k}|=k+2$ and operator $D$ have weight
$|D|=1$. The variable $u_{0}$ will be often denoted as $u$. The derivation $D$
can be represented in the form

$D=X_{u_{1}}=\sum_{k=0}^{\infty}u_{k+1}\frac{\partial}{\partial u_{k}}.$

Derivations of ${\mathfrak{A}}_{0}$ form a Lie algebra ${\rm
Der\,}{\mathfrak{A}}_{0}$ over ${\mathbb{C}}$. A formal sum

$X=\sum_{n=0}^{\infty}f_{n}\frac{\partial}{\partial u_{n}},\qquad
f_{n}\in{\mathfrak{A}}_{0},$ (4)

is a derivation in ${\mathfrak{A}}_{0}$. Its action
$X:{\mathfrak{A}}_{0}\mapsto{\mathfrak{A}}_{0}$ is well defined, since any
element $a\in{\mathfrak{A}}_{0}$ depends on a finite subset of variables, and
therefore the sum $X(a)$ contains only a finite number of non-vanishing terms.
The $\mathbb{C}$ linearity and the Leibniz rule are obviously satisfied. For
example, partial derivatives $\frac{\partial}{\partial u_{i}},\ i=0,1,\ldots$,
are commuting derivations in ${\mathfrak{A}}_{0}$.

A derivation $X$ is said to be evolutionary if it commutes with the derivation
$D$. For an evolutionary derivation it follows from the condition $XD=DX$ that
all coefficients $f_{n}$ in (4) can be expressed as $f_{n}=D^{n}(f)$ in terms
of one element $f\in{\mathfrak{A}}_{0}$, which is called the characteristic of
the evolutionary derivation. We will use notation

$X_{f}=\sum_{i=0}^{\infty}D^{i}(f)\frac{\partial}{\partial u_{i}}$ (5)

for the evolutionary derivation corresponding to the characteristic $f$. The
derivation $D$ is also evolutionary $D=X_{u_{1}}$ with the characteristic
$u_{1}$.

Evolutionary derivations form a Lie subalgebra of the Lie algebra ${\rm
Der\,}{\mathfrak{A}}_{0}$. Indeed,

$\begin{array}[]{l}\alpha X_{f}+\beta X_{g}=X_{\alpha f+\beta
g},\quad\alpha,\beta\in\mathbb{C},\\\
\phantom{}[X_{f},X_{g}]=X_{[f,g]},\end{array}$

where $[f,g]\in{\mathfrak{A}}_{0}$ denotes the Lie bracket

$[f,g]=X_{f}(g)-X_{g}(f),$ (6)

which is bi-linear, skew-symmetric and satisfying the Jacobi identity. Thus
${\mathfrak{A}}_{0}$ is a Lie algebra with Lie bracket defined by (6).

Let $a(u,\ldots,u_{n})$ be a non-constant element of ${\mathfrak{A}}_{0}$.
Then $X_{f}(a)$ can be represented by a finite sum

$X_{f}(a)=\sum_{i=0}^{n}\frac{\partial a}{\partial u_{i}}D^{i}(f)=a_{*}(f),$
(7)

where

$a_{*}=\sum_{i=0}^{n}\frac{\partial a}{\partial u_{i}}D^{i}$ (8)

is the Fréchet derivative of $a(u,\ldots,u_{n})$ and $a_{*}(f)$ is the Fréchet
derivative of $a$ in the direction $f$. Using the Fréchet derivative we can
represent the Lie bracket (6) in the form

$[f\,,\,g]=g_{*}(f)-f_{*}(g).$ (9)

An evolutionary derivation $X_{f}$ we identify with the partial differential
equation

$\partial_{t}(u)=f,\qquad f\in{\mathfrak{A}}_{0}.$ (10)

Following [28] we define symmetries of (10).

######  Definition 1.

A dynamical system

$\partial_{\tau}(u)=g,\qquad g\in{\mathfrak{A}}_{0}$ (11)

is called an inﬁnitesimal symmetry (or just symmetry for brevity) for (10) if
(10) and (11) are compatible.

It is clear that equation (11) is a symmetry of equation (10) iff
$[X_{f},X_{g}]=0$. By a symmetry we will also call the evolutionary derivation
$\partial_{\tau}$ which commutes with $\partial_{t}$.

###  1.2. Cyclic Frobenius algebras.

Let $\mathcal{B}$ be an associative $\mathbb{C}$-algebra with the unit 1, and
let $\mathcal{M}$ be some $\mathbb{C}$-linear space, $\dim\mathcal{M}\geqslant
1$.

######  Definition 2.

A cyclic Frobenius algebra ($\rm{CF}$-algebra) $\mathcal{B}$ is an algebra
$\mathcal{B}$ with a $\mathbb{C}$-bilinear skew-symmetric form
$\Phi(\cdot,\cdot)\colon\mathcal{B}\otimes_{\mathbb{C}}\mathcal{B}\to\mathcal{M}$
such that for any $A,B,C\in\mathcal{B}$

$\Phi(A,BC)+\Phi(B,CA)+\Phi(C,AB)=0.$ (12)

Examples:
1) $\mathcal{M}=\mathcal{B}$ and $\Phi(A,B)=AB-BA=[A,B]$;
2) $\mathcal{B}$ is a commutative algebra with Poisson bracket
$\\{\cdot,\cdot\\}$, $\mathcal{M}=\mathcal{B}$ and $\Phi(A,B)=\\{A,B\\}$;
3) $\rm{CF}$-algebras, where $\mathcal{M}$ is a field, obtained from the
constructions of algebras from [2], [25].

Let $\mathcal{K}=\\{B\in\mathcal{B}\,:\,\Phi(A,B)=0,\text{ for
all}\,A\in\mathcal{B}\\}$. From Definition 2 it follows:

######  Lemma 3.

The annililator $\mathcal{K}$ of the form $\Phi$ is the subring in
$\mathcal{B}$, $1\in\mathcal{K}$, and

$\Phi(A,bC)=\Phi(Ab,C)\;\text{ for all }A,C\in\mathcal{B}\;\text{ and
}b\in\mathcal{K}.$

We shall assume that $u$ is a smooth function
$u=u(t_{1},t_{3},\ldots,t_{2k-1},\ldots)$ of graded variables
$t_{2k-1},\,k=1,2,\ldots$, where $|t_{2k-1}|=1-2k$. The variable $t_{1}$ we
will identity with $x$. We use abbreviated notations for partial derivatives
$\frac{\partial u}{\partial t_{2k-1}}=\partial_{t_{2k-1}}(u)$ and
$\partial_{x}=\partial_{t_{1}}=D$. The grading weights
$|\partial_{t_{2k-1}}|=2k-1$.

Let us define a differential operator of order $m$ as a finite sum of the form

$A=\sum\limits_{i=0}^{m}a_{i}D^{i},\;a_{i}\in{\mathfrak{A}}_{0},\,a_{m}\neq 0$

where $D^{0}=1$ is the identity operator.

An operator $A$ is called _homogeneous_ of weight $k$, if $|a_{i}|+i=k$ for
all $i$. Differential operators act naturally on the algebra
${\mathfrak{A}}_{0}$.

The set of differental graded operators

${\mathfrak{A}}_{0}[D]=\Big{\\{}\sum\limits_{i=0}^{m}a_{i}D^{i}\,|\,a_{i}\in{\mathfrak{A}}_{0},\,a_{m}\neq
0,\,m\in\mathbb{Z}_{\geqslant 0}\Big{\\}}$

and the set of graded formal differential series

${\mathfrak{A}}_{0}^{D}={\mathfrak{A}}_{0}[D][[D^{-1}]]=\Big{\\{}\sum\limits_{i\leqslant
m}a_{i}D^{i}\,|\,a_{i}\in{\mathfrak{A}}_{0},\,a_{m}\neq
0,\,m\in{\mathbb{Z}}\Big{\\}}$ (13)

are non-commutative associative algebras. In this algebra, multiplication is
defined by the composition of series using the formula

$bD^{k}a\,D^{l}=\sum_{i\geqslant 0}\binom{k}{i}ba^{(i)}D^{k+l-i}$ (14)

reflecting the Leibniz rule. Here $a^{(i)}=D^{i}(a)\in{\mathfrak{A}}_{0}$ and

$\binom{k}{0}=1,\quad\binom{k}{i}=\frac{k(k-1)\cdots(k-i+1)}{i!}=(-1)^{i}\binom{-k+i-1}{i},\;i>0,\;k\in\mathbb{Z}.$
(15)

Obviously ${\mathfrak{A}}_{0}[D]\subset{\mathfrak{A}}_{0}^{D}$ and
${\mathfrak{A}}_{0}[D]$ is a subalgebra in ${\mathfrak{A}}_{0}^{D}$.

For example,

$\displaystyle D^{k}a$
$\displaystyle=\,\sum_{i=0}^{k}\binom{k}{i}a^{(i)}D^{k-i},\,k\geqslant 0;$
$\displaystyle D^{-1}a$ $\displaystyle=\,\sum_{i\geqslant
0}(-1)^{i}a^{(i)}D^{-(i+1)}=aD^{-1}-D(a)D^{-2}+D^{2}(a)D^{-3}-D^{3}(a)D^{-4}+\cdots$

For any two elements $A,B\in{\mathfrak{A}}_{0}^{D}$ we have the commutator
$[A,B]=AB-BA$. For instance, for any $a\in{\mathfrak{A}}_{0}$, the formulas
are fulfilled: $[D,a]=D(a)$; $[D^{-1},a]=-D(a)D^{-2}+D^{2}(a)D^{-3}-\cdots$

######  Definition 4.

For a formal series $A\in{\mathfrak{A}}_{0}^{D}$ the coefficient $a_{-1}$ of
the term $a_{-1}D^{-1}$ is called the residue of this series $A$ and denoted
by ${\rm res}\,A$.

######  Lemma 5.

  1. 1)

For any $B\in{\mathfrak{A}}_{0}^{D}$ and $a\in{\mathfrak{A}}_{0}$ we have
${\rm res}\,[a,B]=0$.

  2. 2)

For any $a\in{\mathfrak{A}}_{0}$ and $B,C\in{\mathfrak{A}}_{0}^{D}$

${\rm res}\,[aB,C]={\rm res}\,[B,Ca].$ (16)

###### Proof.

1) Let $B=\sum\limits_{k\leqslant m}b_{k}D^{k}$. Then

$[a,B]=-\sum_{k\leqslant m}\sum\limits_{i>0}\binom{k}{i}b_{k}a^{(i)}D^{k-i}.$

Therefore

${\rm res}\,[a,B]=\binom{k}{k+1}b_{k}a^{(k+1)}=0,\quad k+1>0.$

2) For any elements $A,B,C$ of any associative algebra, the identity

$[A,BC]+[B,CA]+[C,AB]=0$ (17)

holds. Therefore, for $a\in{\mathfrak{A}}_{0}$ and
$B,C\in{\mathfrak{A}}_{0}^{D}$

$[aB,C]=[B,Ca]+[a,BC].$ (18)

Applying the operator “${\rm res}$” to (18) and using already proved statement
1), we obtain the proof of statement 2). ∎

The monomials

$u^{\xi}=u_{n}^{i_{n}}\cdots
u_{0}^{i_{0}},\;\xi=(i_{n},\ldots,i_{0}),\;i_{n}>0,\;i_{k}\geqslant
0,\;k=0,\ldots,n-1,\;|u^{\xi}|=\sum\limits_{k=0}^{n}(k+2)i_{k},$

form an additive basis of the graded algebra
${\mathfrak{A}}_{0}=\mathbb{C}[u_{0},u_{1},\ldots]$. We will consider
${\mathfrak{A}}_{0}$ as a graded algebra
${\mathfrak{A}}_{0}=\mathbb{C}\oplus\widetilde{{\mathfrak{A}}}_{0}$, where
$\widetilde{{\mathfrak{A}}}_{0}=\oplus_{m}{\mathfrak{A}}_{0}^{m}$ and
${\mathfrak{A}}_{0}^{m}$ is a graded finite-dimensional $\mathbb{C}$-linear
subspace in ${\mathfrak{A}}_{0}$ with an additive basis
$\\{u^{\xi},\,|u^{\xi}|=m\\}$.

For example, $\\{u\\},\,\\{u_{1}\\}$, and $\\{u_{2},u^{2}\\}$ are the basises
of the spaces ${\mathfrak{A}}_{0}^{2},\,{\mathfrak{A}}_{0}^{3}$, and
${\mathfrak{A}}_{0}^{4}$, respectively.

The vectors of the space ${\mathfrak{A}}_{0}^{m}$ are called homogeneous
polynomials of weight $m$. Let us introduce an ordering of the multiplicative
generators of the algebra ${\mathfrak{A}}_{0}$:

$u=u_{0}<u_{1}<\cdots<u_{k}<u_{k+1}<\cdots$

Then a strict order is defined in the monomial basis $\\{u^{\xi}\\}$ of the
space ${\mathfrak{A}}_{0}^{m}$ for each $m>0$. This order is induced by the
lexicographic order of the sequences $\xi$.

######  Lemma 6.

For any $m>0$ the homomorphism
$D\colon{\mathfrak{A}}_{0}\to{\mathfrak{A}}_{0}$ defines a monomorphism
${\mathfrak{A}}_{0}^{m}\to{\mathfrak{A}}_{0}^{m+1}$.

###### Proof.

By definition, $D(1)=0$ and $D(u_{k})=u_{k+1},\,k=0,1,\ldots$ Therefore, the
differentiation operator $D$ takes ${\mathfrak{A}}_{0}^{m}$ to
${\mathfrak{A}}_{0}^{m+1}$. Let $u^{\xi}\in{\mathfrak{A}}_{0}^{m}$ where
$\xi=(i_{n},\ldots,i_{0}),\;i_{n}\neq 0$. Then
$D(u^{\xi})=i_{n}u_{n+1}u_{n}^{i_{n}-1}u^{\hat{\xi}}+u_{n}^{i_{n}}D(u^{\hat{\xi}})$.
The composition of linear homomorphisms

$\overline{D}\colon{\mathfrak{A}}_{0}^{m}\to{\mathfrak{A}}_{0}^{m+1}\to{\mathfrak{A}}_{0}^{m+1}\;:\;\overline{D}(u^{\xi})=u^{\xi^{\prime}}=i_{n}u_{n+1}u_{n}^{i_{n}-1}u^{\hat{\xi}}$

maps the ordered set of monomials $u^{\xi}\in{\mathfrak{A}}_{0}^{m}$ into the
ordered set of monomials $u^{\xi^{\prime}}\in{\mathfrak{A}}_{0}^{m+1}$. We
will show that this mapping is monotone and thus we obtain that the
homomorphism $D$ is a monomorphism for $m>0$.

Let
$\xi_{1}=(i_{n_{1}},\ldots,i_{0_{1}})>\xi_{2}=(i_{n_{2}},\ldots,i_{0_{2}})$
where $i_{n_{1}}\neq 0$ and $i_{n_{2}}\neq 0$. Then $n_{1}\geqslant n_{2}$. If
$n_{1}>n_{2}$ then $\xi_{1}^{\prime}>\xi_{2}^{\prime}$. If $n_{1}=n_{2}$ and
$i_{n_{1}}>i_{n_{2}}$ then in this case it is also obvious that
$\xi_{1}^{\prime}>\xi_{2}^{\prime}$. Finally, let $n_{1}=n_{2}=n$ and
$i_{n_{1}}=i_{n_{2}}=i_{n}$. Then there is a sequence
$\zeta=(i_{n},\ldots,i_{k}),\;0<k\leqslant n$, such that
$\xi_{1}=(\zeta,\eta_{1})$ and $\xi_{2}=(\zeta,\eta_{2})$ where
$\eta_{1}>\eta_{2}$. In this case
$u^{\xi_{1}^{\prime}}=\overline{D}(u^{\xi_{1}})=\overline{D}(u^{\zeta})u^{\eta_{1}}$
and
$u^{\xi_{2}^{\prime}}=\overline{D}(u^{\xi_{2}})=\overline{D}(u^{\zeta})u^{\eta_{2}}$
and therefore $u^{\xi_{1}^{\prime}}>u^{\xi_{2}^{\prime}}$. Lemma 6 is proved.
∎

We introduce a homogeneous $\mathbb{C}$-bilinear skew-symmetric form

$\sigma(\cdot,\cdot)\colon{\mathfrak{A}}_{0}^{D}\otimes{\mathfrak{A}}_{0}^{D}\to{\mathfrak{A}}_{0},\;|\sigma(A,B)|=|A|+|B|$

by formula

$\sigma(aD^{n},bD^{m})=\begin{cases}\binom{n}{n+m+1}\sum\limits_{s=0}^{n+m}(-1)^{s}a^{(s)}b^{(n+m-s)},&\text{if
}\,n+m\geqslant 0,\,nm<0,\\\ 0,&otherwise.\end{cases}$ (19)

Using formula (19), we obtain:

######  Theorem 7.

  1. 1)

For any $A,B\in{\mathfrak{A}}_{0}^{D}$

$D\big{(}\sigma(A,B)\big{)}={\rm res}\,[A,B].$ (20)

  2. 2)

For any $A\in{\mathfrak{A}}_{0}^{D}$ we have

$\sigma(D,A)={\rm res}\,A.$ (21)

######  Corollary 8.

For any $A,B\in{\mathfrak{A}}_{0}^{D}$ we have

$\sigma\big{(}D,[A,B]\big{)}=D\big{(}\sigma(A,B)\big{)}.$ (22)

######  Theorem 9.

The algebra ${\mathfrak{A}}_{0}^{D}$ is a $\rm{CF}$-algebra with the form
$\Phi(\cdot,\cdot)=\sigma(\cdot,\cdot)$ and $\mathcal{M}={\mathfrak{A}}_{0}$.

###### Proof.

Let $A,B,C\in{\mathfrak{A}}_{0}^{D}$. Take the residue ${\rm res}$ of the left
side of equality (17). Then, according to (20), we obtain:

$D\big{(}\sigma(A,BC)+\sigma(B,CA)+\sigma(C,AB)\big{)}=0.$

Since according to Lemma 6 the operator $D$ is the monomorphism on non-
constant series, we obtain that relation (12) is true. Theorem 9 is proved. ∎

######  Corollary 10.

  1. 1)

The annililator $\mathcal{K}$ of the form $\sigma(\cdot,\cdot)$ is
${\mathfrak{A}}_{0}$.

  2. 2)

On the set of generators $\\{D^{n},\,n\in\mathbb{Z}\\}$ of the free left
${\mathfrak{A}}$-module ${\mathfrak{A}}^{D}$, the form $\sigma(\cdot,\cdot)$
is non-degenerate.

The proof of this corollary is obtained directly from the previous definitions
and results.

For $A=\sum\limits_{i\leqslant m}a_{i}D^{i},\,a_{m}\neq 0$, we put
$A=A_{+}+A_{-}$ where $A_{+}=0$ if $m<0$, and
$A_{+}=\sum\limits_{i=0}^{m}a_{i}D^{i}$ if $m\geqslant 0$.

######  Corollary 11.

  1. 1)

$\sigma(A,B)=\sigma(A_{+},B_{-})+\sigma(A_{-},B_{+})$.

  2. 2)

Let $[A,B]=0$. Then $\sigma(A,B)=0$.

###### Proof.

Assertion 1) follows from formula (19). Assertion 2) follows from Corollary 8
and Lemma 6. ∎

######  Theorem 12.

The form $\sigma(\cdot,\cdot)$ is given in terms of the operation ${\rm res}$
by the recursive formula

$\sigma(A,BD)=\sigma(DA,B)-{\rm res}\,AB$ (23)

with the initial condition $\sigma(A,bD)=-{\rm res}\,Ab$ for any
$A\in{\mathfrak{A}}_{0}^{D}$ and $b\in{\mathfrak{A}}_{0}$.

###### Proof.

As was noted above, ${\rm res}\,A=\sigma(D,A)$. For the triple $(A,B,D)$,
according to identity (12), we obtain formula (23). ∎

For example:

$\sigma(A,bD^{2})=\sigma(DA,bD)-{\rm res}\,AbD=-{\rm res}\,(DAb+AbD)$

for any $A\in{\mathfrak{A}}_{0}^{D}$ and $b\in{\mathfrak{A}}_{0}$.

###  1.3. KdV hierarchy.

Let us consider a homogeneous operator $L=D^{2}-u$, $|L|=2$.

######  Lemma 13.

A homogeneous formal series

${\mathcal{L}}=D+\sum_{n\geqslant 1}I_{1,n}D^{-n},\;|{\mathcal{L}}|=1,$

where $I_{1,n}\in{\mathfrak{A}}_{0}$ are homogeneous polynomials of the weight
$n+1$, satisfies the equation ${\mathcal{L}}^{2}=L$ if and only if
$I_{1,1}=-\frac{1}{2}u,\;I_{1,2}=\frac{1}{4}u_{1}$ and

$2I_{1,n}+I^{\prime}_{1,n-1}+\sum_{k=1}^{n-2}I_{1,k}\sum_{i=0}^{n-k-2}(-1)^{i}\binom{k+i-1}{i}I^{(i)}_{1,n-k-i-1}=0,\;n\geqslant
3.$ (24)

###### Proof.

Consider the equation

$\left(D+\sum_{k\geqslant 1}I_{1,k}D^{-k}\right)\left(D+\sum_{q\geqslant
1}I_{1,q}D^{-q}\right)=D^{2}-u.$

We obtain

$\sum_{q\geqslant 1}DI_{1,q}D^{-q}+\sum_{k\geqslant
1}I_{1,k}D^{-k+1}+\sum_{k\geqslant 1,q\geqslant
1}I_{1,k}D^{-k}I_{1,q}D^{-q}=-u.$

Using (14), we get (24). ∎

Formula (24) allows to calculate the polynomials $I_{1,n},\,n\geqslant 3$,
recursively

${\mathcal{L}}=D-\frac{1}{2}uD^{-1}+\frac{1}{4}u_{1}D^{-2}-\frac{1}{8}(u_{2}+u^{2})D^{-3}+\frac{1}{16}(u_{3}+6uu_{1})D^{-4}-\\\
-\frac{1}{32}(u_{4}+14u_{2}u+11u_{1}^{2}+2u^{3})D^{-5}+\ldots$

Let us define a sequence of differential operators

$A_{2k-1}={\mathcal{L}}_{+}^{2k-1}=D^{2k-1}-\frac{1}{2}(2k-1)uD^{2k-3}+\cdots+a_{2k-1},\,k=1,2,\ldots,$
(25)

where $a_{2k-1}=A_{2k-1}(1)\in{\mathfrak{A}}_{0},\,|a_{2k-1}|=2k-1$, and
homogeneous differential polynomials
$\rho_{2k}\in{\mathfrak{A}}_{0},\;|\rho_{2k}|=2k$,

$\rho_{0}=1,\quad\rho_{2k}={\rm res}\,{\mathcal{L}}^{2k-1},\;k=1,2,\ldots$
(26)

Thus

${\mathcal{L}}^{2k-1}=A_{2k-1}+\rho_{2k}D^{-1}+\ldots,\;k>0.$

We have $a_{1}=0$ and $\rho_{2}=-\frac{1}{2}u$.

Let be

${\mathcal{L}}^{2k-1}=A_{2k-1}+\sum\limits_{n\geqslant
1}I_{2k-1,n}D^{-n},\;k>0.$

From the relation ${\mathcal{L}}^{2k+1}=L{\mathcal{L}}^{2k-1}$ we obtain

$\displaystyle A_{2k+1}$
$\displaystyle=(D^{2}-u)A_{2k-1}+I_{2k-1,1}D+(I_{2k-1,2}+2I^{\prime}_{2k-1,1}),$
(27) $\displaystyle I_{2k+1,n}$
$\displaystyle=I_{2k-1,n+2}+2I^{\prime}_{2k-1,n+1}+I^{\prime\prime}_{2k-1,n}-uI_{2k-1,n}.$
(28)

######  Corollary 14.

$a_{2k+1}=(D^{2}-u)(a_{2k-1})+I_{2k-1,2}+2\rho^{\prime}_{2k},\,k\geqslant 1,$
(29)
$\rho_{2k+2}=I_{2k-1,3}+2I^{\prime}_{2k-1,2}+\rho^{\prime\prime}_{2k}-u\rho_{2k}.$
(30)

Let $J=\langle u,u_{1},\ldots\rangle\subset{\mathfrak{A}}_{0}$ be the two-
sided maximal ideal generated by $u,u_{1},\ldots$

######  Proposition 15.

For $k\in\mathbb{N}$ the following formula holds:

$\rho_{2k+2}=-\frac{1}{2^{2k+1}}\left(u_{2k}+\ldots+(-1)^{k}\binom{2k+1}{k}u^{k+1}\right)=-\frac{1}{2^{2k+1}}(u_{2k}-\widehat{\rho}_{2k+2})$
(31)

where $\widehat{\rho}_{2k+2}\in J^{2}$.

###### Proof.

By definition, $\rho_{2n}={\rm res}\,L^{\frac{2n-1}{2}},\,n\geqslant 1$. Since
$[D,u]=u_{1}$, then to calculate the coefficient at $u^{n}$, it is sufficient
to calculate the coefficient at $x^{-1}$ of the series
$f(x)=(x^{2}-a)^{\frac{2n-1}{2}}$ where $[x,a]=0$. We have

$f(x)=x^{2n-1}\Big{(}1-x^{-2}a\Big{)}^{\frac{2n-1}{2}}=x^{2n-1}\left(1+\sum_{k\geqslant
1}(-1)^{k}\binom{\frac{2n-1}{2}}{k}x^{-2k}a^{k}\right).$

Therefore, the desired coefficient is

$(-1)^{n}\binom{\frac{2n-1}{2}}{n}a^{n}=(-1)^{n}\frac{1}{2^{2n-1}}\binom{2n-1}{n}a^{n}.$

Formula (24) implies that

$I_{1,n}=(-1)^{n}\frac{1}{2^{n}}u_{n-1}\\!\mod J^{2},\;n>0.$

Using formulas (28) and (30), we obtain by induction that

$\rho_{2k+2}=-\frac{1}{2^{2k+1}}u_{2k}\\!\mod J^{2},\;k\geqslant 0.$

∎

Examples:

$\displaystyle\rho_{4}$
$\displaystyle=-\frac{1}{8}(u_{2}-3u^{2}),\qquad\rho_{6}=-\frac{1}{32}(u_{4}-10u_{2}u-5u_{1}^{2}+10u^{3}),$
$\displaystyle\rho_{8}$
$\displaystyle=-\frac{1}{128}(u_{6}-28u_{3}u_{1}-14u_{4}u-21u_{2}^{2}+70u_{2}u^{2}+70u_{1}^{2}u-35u^{4}).$

It is easy to show that $[{\mathcal{L}}^{2k-1},L]=0$ and therefore the
commutator

$[A_{2k-1},L]=[{\mathcal{L}}^{2k-1}-({\mathcal{L}}^{2k-1})_{-},L]=2D(\rho_{2k})\in{\mathfrak{A}}_{0},$

is the operator of multiplication on the function $2D(\rho_{2k})$.

######  Definition 16.

The KdV hierarchy is defined as an infinite sequence of differential equations

$\partial_{t_{2k-1}}(u)=-2D(\rho_{2k}),\,k\in\mathbb{N}.$ (32)

Examples:

$\begin{array}[]{rll}\partial_{t_{1}}(u)&=&u_{1},\\\
4\partial_{t_{3}}(u)&=&u_{3}-6uu_{1},\\\
16\partial_{t_{5}}(u)&=&u_{5}-10uu_{3}-20u_{1}u_{2}+30u^{2}u_{1},\end{array}$

and so on.

The partial derivatives $\partial_{t_{2k-1}}$ can be extended to derivations
of the algebra ${\mathfrak{A}}_{0}^{D}$

$\partial_{t_{2k-1}}(A)=\sum_{i\leqslant
m}\partial_{t_{2k-1}}(a_{i})D^{i},\;\text{ where }A=\sum_{i\leqslant
m}a_{i}D^{i}.$

Therefore the KdV hierarchy can be written in the form of Lax’s equations

$\partial_{t_{2k-1}}(L)=[A_{2k-1},L].$ (33)

It can be shown, that the derivations $\partial_{t_{2k-1}}$ commute with each
other [28], and thus the KdV hierarchy is a system of compatible equations.

It follows from
$\partial_{t_{2k-1}}(L)=\partial_{t_{2k-1}}({\mathcal{L}}^{2})=\partial_{t_{2k-1}}({\mathcal{L}}){\mathcal{L}}+{\mathcal{L}}\partial_{t_{2k-1}}({\mathcal{L}})$
and (33) that

$\partial_{t_{2k-1}}({\mathcal{L}})=[A_{2k-1},{\mathcal{L}}],$

and therefore,

$\partial_{t_{2k-1}}({\mathcal{L}}^{2n-1})=[A_{2k-1},{\mathcal{L}}^{2n-1}],\;n,k\in\mathbb{N}.$
(34)

Let’s put

$\sigma_{2k-1,2n-1}=\sigma({\mathcal{L}}^{2k-1}_{+},{\mathcal{L}}^{2n-1}_{-})\in{\mathfrak{A}}_{0}.$
(35)

According to Corollary 11, we obtain:

$\sigma_{2k-1,2n-1}=\sigma_{2n-1,2k-1}.$ (36)

Taking the residue from the equation (34), we get

$\partial_{t_{2k-1}}(\rho_{2n})=D(\sigma_{2k-1,2n-1}).$ (37)

Thus $\\{\rho_{2n},\ n\in\mathbb{N}\\}$ is a sequence of common conserved
densities for the infinite KdV hierarchy (32), and $\sigma_{2k-1,2n-1}$ are
homogeneous $D$-differential polynomials from $u$,
$|\sigma_{2k-1,2n-1}|=2n+2k-2$.

On the algebra ${\mathfrak{A}}_{0}$ the evolutionary derivations
$\partial_{t_{2k-1}}$ are represented by commuting derivations

$D_{2k-1}=-2\sum_{\ell=0}^{\infty}D^{\ell+1}(\rho_{2k})\frac{\partial}{\partial
u_{\ell}}.$ (38)

In particular $D_{1}=D$,

$D_{3}=\frac{1}{4}(u_{3}-6uu_{1})\frac{\partial}{\partial
u}+\frac{1}{4}(u_{4}-6uu_{2}-6u_{1}^{2})\frac{\partial}{\partial
u_{1}}+\cdots$

###  1.4. The $N$-th Novikov hierarchy.

Let us choose a positive integer $N$. Let
$\mathcal{A}=\mathbb{C}[\alpha_{4},\ldots,\alpha_{2N+2}]$ be a graded algebra
of parameters and ${\mathfrak{A}}=\mathcal{A}[u_{0},u_{1},\ldots]$. We assume
that $|\alpha_{2n}|=2n,\,n\geqslant 2$, and $\alpha_{2n}$ are constants,
meaning that $D_{2k-1}(\alpha_{2n})=0$ for all $k\geqslant 1$ and $n\geqslant
2$.

Let us define a symmetry $\partial_{\tau}$ of the KdV equation taking a linear
combination with constant coefficients of the first $N$ members of the KdV
hierarchy (32)

$\partial_{\tau}(u)=\partial_{t_{2N+1}}(u)+\sum_{k=1}^{N-1}\alpha_{2(N-k+1)}\partial_{t_{2k-1}}(u).$
(39)

Let us define a polynomial

$F_{2N+2}=\rho_{2N+2}+\sum_{k=0}^{N-1}\alpha_{2(N-k+1)}\rho_{2k}.$ (40)

In (40) we assume that $\rho_{0}=1$ and $\alpha_{2N+2}$ is a constant
parameter of weight $|\alpha_{2N+2}|=2N+2$. The polynomial $F_{2N+2}$ (see
(40)) is homogeneous of weight $2N+2$. Let us restrict ourselves with
solutions of the KdV hierarchy which are invariant with respect to the
symmetry (39). It implies that

$\rho_{2N+2}+\sum_{k=0}^{N-1}\alpha_{2(N-k+1)}\rho_{2k}=0.$ (41)

It follows from (31) that equation (41) can be resolved with respect to the
variable $u_{2N}$ and written in the form

$u_{2N}=f_{2N+2}(u_{0},u_{1},\ldots,u_{2N-2})$ (42)

where
$f_{2N+2}=\widehat{\rho}_{2N+2}+2^{2N+1}\sum\limits_{k=0}^{N-1}\alpha_{2(N-k+1)}\rho_{2k}\in{\mathfrak{A}}$
is a homogeneous polynomial, $|f_{2N+2}|=2N+2$. Equation (42) is called $N$-th
Novikov equation. Since $\rho_{2n}\in{\mathfrak{A}}_{0}$, these equations
depend linearly on $\alpha_{4},\ldots,\alpha_{2N+2}$.

For example:

$\begin{array}[]{ll}N=1:\\!\\!\\!&u_{2}=3u^{2}+8\alpha_{4},\\\
N=2:\\!\\!\\!&u_{4}=10(u_{2}-u^{2})u+5u_{1}^{2}-16\alpha_{4}u+32\alpha_{6},\\\
N=3:\\!\\!\\!&u_{6}=14(u_{4}\\!-\\!5u_{2}u\\!+\\!5u_{1}^{2})u+28u_{1}u_{3}+21u_{2}^{2}+35u^{4}-16\alpha_{4}(u_{2}\\!-\\!3u^{2})-64\alpha_{6}u+128\alpha_{8}.\end{array}$

Since $u_{k}=D^{k}(u)=u^{(k)}$, the $N$-th Novikov equation is an ordinary
differential equation of the $2N$-th order for the function $u=u(x)$.

Let ${\mathfrak{I}}_{N}=(F_{2N+2})\subset{\mathfrak{A}}$ be a differential
ideal generated by the polynomial $F_{2N+2}$ and the $D$-derivatives of
$F_{2N+2}$. For any element of ${\mathfrak{A}}$ the canonical projection

$\pi_{N}\,:\,{\mathfrak{A}}\mapsto{\mathfrak{A}}\diagup{\mathfrak{I}}_{N}$

is the result of the elimination of variables $u_{k},\ k\geqslant 2N$, using
equation (42) and equation $u_{2N+k}=D^{k}(f_{2N+2})$ recursively.

######  Proposition 17.

The ideal ${\mathfrak{I}}_{N}$ is invariant with respect to evolutionary
derivations $\partial_{t_{2k-1}},\,k\in\mathbb{N}$.

###### Proof.

Indeed, it follows from (7), (36), and (38) that

$\partial_{t_{2k-1}}(F_{2N+2})=\partial_{t_{2k-1}}\Big{(}\rho_{2N+2}+\sum\limits_{\ell=0}^{N-1}\alpha_{2(N-\ell+1)}\rho_{2\ell}\Big{)}=\\\
=\Big{(}\partial_{2N+1}+\sum\limits_{\ell=1}^{N-1}\alpha_{2(N-\ell+1)}\partial_{2\ell-1}\Big{)}(\rho_{2k})=-2(\rho_{2k})_{*}(D(F_{2N+2}))\subset{\mathfrak{I}}_{N}.$
(43)

∎

Commuting derivations $D_{2k-1},\,1\leqslant k\leqslant N$ (see (38)) induce
on ${\mathfrak{A}}/{\mathfrak{I}}_{N}$ the derivations

$\displaystyle{\mathcal{D}}=$
$\displaystyle\;{\mathcal{D}}_{1}=\sum\limits_{\ell=0}^{2N-2}u_{\ell+1}\dfrac{\partial}{\partial
u_{\ell}}+f_{2N}\dfrac{\partial}{\partial u_{2N-1}}\,,$
$\displaystyle{\mathcal{D}}_{2k-1}=$
$\displaystyle\,-2\sum\limits_{\ell=0}^{2N-1}{\mathcal{D}}^{\ell+1}(\rho_{2k})\dfrac{\partial}{\partial
u_{\ell}},\quad 1\leqslant k\leqslant N.$

In $\mathbb{C}^{2N}$ there are $N$ compatible systems of $N$ ordinary
differential equations

$\partial_{t_{2k-1}}(u_{s})={\mathcal{D}}_{{2k-1}}(u_{s})=-2{\mathcal{D}}^{s+1}(\rho_{2k}),\quad
s=0,\ldots,N-1,\;k=1,\ldots,N,$ (44)

which we will call $N$-th Novikov hierarchy. In this case, the parameters
$\alpha_{2k}$ are assumed to be fixed complex numbers. In the hierarchy (44),
system with $k=1,\,s=0,\ldots,N-1$ represents the $N$-th Novikov equation (42)
as a first order system of $2N$ ordinary differential equations.

######  Proposition 18.

The $N$-th Novikov equation possesses $N$ first integrals

$H_{2n+1,2N+1}=\sigma_{2n+1,2N+1}+\sum_{k=1}^{N-1}\alpha_{2N-2k+2}\sigma_{2n+1,2k-1},\quad
n=1,\ldots,N.$ (45)

The polynomials $H_{2n+1,2N+1}$ are homogeneous of weight
$|H_{2n+1,2N+1}|=2N+2n+2$.

###### Proof.

It follows from (36) that

$\partial_{t_{2n-1}}(F_{2N+2})=D(H_{2n+1,2N+1})$

where

$H_{2n+1,2N+1}=\sigma_{2n+1,2N+1}+\sum_{k=1}^{N-1}\alpha_{2N-2k+2}\sigma_{2n+1,2k-1}.$
(46)

Thus, it follows from (43) that
$D(H_{2n+1,2N+1})=\partial_{t_{2n+1}}(F_{2N+2})\in{\mathfrak{I}}_{N}$ and thus
vanish in ${\mathfrak{A}}/{\mathfrak{I}}_{N}$. For $n=1,\ldots,N$ one can
check that $H_{2n+1,2N+1}\not\in{\mathfrak{I}}_{N}$, and thus $H_{2n+1,2N+1}$
is a first integral of Novikov equation. Moreover the first integrals
$H_{2n+1,2N+1},\ n=1,\ldots,N$, are algebraically independent. ∎

It is known that $N$-th Novikov equation and equations of the $N$-th Novikov
hierarchy can be reduced to integrable Hamiltonian systems [6]. It follows
from the general theory of integrable Hamiltonian systems that the derivations
${\mathcal{D}}_{t_{2s-1}}$ with $s>N$ in ${\mathfrak{A}}/{\mathfrak{I}}_{N}$
are dependent, since they are linear combinations of
${\mathcal{D}}_{t_{2k-1}},\ k=1,\ldots,N$, i.e.

${\mathcal{D}}_{t_{2s-1}}=\sum\limits_{k=1}^{N}a_{k}{\mathcal{D}}_{t_{2k-1}}$
(47)

with coefficients
$a_{k}\in{\mathbb{C}}[\alpha_{4},\ldots,\alpha_{2N+2},H_{2N+1,3},\ldots,H_{2N+1,2N+1}]$
where $H_{2n+1,2N+1}$, $n=1,\ldots,N$, are first integrals of the $N$-th
Novikov equation. Using (45) one can find the polynomials $H_{2n+1,2N+1},\
n>N$, which are also first integrals (it follows from the proof of Proposition
18), but they are algebraically dependent with
$H_{3,2N+1},\ldots,H_{2N+1,2N+1}$.

For example:

${\bf N=1}$: The $N=1$ Novikov equation coincides with the Newton equation
$\partial_{t_{1}}^{2}u=3u^{2}+8\alpha_{4}$.

$\displaystyle\partial_{t_{1}}(u)$ $\displaystyle=\,u_{1};$
$\displaystyle\partial_{t_{1}}(u_{1})$ $\displaystyle=\,3u^{2}+8\alpha_{4},$

and according Proposition 18 we get one first integral

$H_{3,3}=-\frac{3}{16}\left(\frac{1}{2}u_{1}^{2}-u^{3}-8\alpha_{4}u\right).$
(48)

${\bf N=2}$: The hierarchy consists of two compatible systems in which the
first one is the $N=2$ Novikov equation

$\displaystyle\partial_{t_{1}}(u_{s})$ $\displaystyle=\,u_{s+1},\quad
s=0,1,2;$ $\displaystyle\partial_{t_{1}}(u_{3})$
$\displaystyle=\,32\alpha_{6}-10u^{3}-16\alpha_{4}u+10u_{2}u+5u_{1}^{2};$
$\displaystyle 4\partial_{t_{3}}(u_{s})$
$\displaystyle=\,{\mathcal{D}}^{s+1}(u_{2}-3u^{2}),\quad s=0,1,2,3\,.$

Proposition 18 give us two first integrals

$\displaystyle H_{3,5}$ $\displaystyle=$
$\displaystyle-\frac{3}{128}\left(5u^{4}+16\alpha_{4}u^{2}-64\alpha_{6}u-10u_{1}^{2}u-u_{2}^{2}+2u_{3}u_{1}\right),$
(49) $\displaystyle H_{5,5}$ $\displaystyle=$
$\displaystyle\frac{5}{512}(24u^{5}+64\alpha_{4}u^{3}-20u_{2}u^{3}-192\alpha_{6}u^{2}-30u_{1}^{2}u^{2}-32\alpha_{4}u_{2}u+$
$\displaystyle+$ $\displaystyle
16\alpha_{4}u_{1}^{2}+64\alpha_{6}u_{2}+4u_{2}^{2}u+12u_{3}u_{1}u-u_{3}^{2}-2u_{2}u_{1}^{2}).$

Obviously $H_{2n+1,2N+1}$ (see (45) in case $N=2$) are first integrals for any
$n$, but they are algebraically dependent with $H_{3,5},H_{5,5}$

$\displaystyle H_{7,5}$
$\displaystyle=\,\frac{7}{6}(3\alpha_{6}^{2}-2\alpha_{4}H_{3,5}),$
$\displaystyle H_{9,5}$
$\displaystyle=\,-3\alpha_{6}H_{3,5}-\frac{9\alpha_{4}}{5}H_{5,5},$
$\displaystyle H_{11,5}$
$\displaystyle=\,\frac{11}{90}\Big{(}-45\alpha_{4}\alpha_{6}^{2}-18\alpha_{6}H_{5,5}+30\alpha_{4}^{2}H_{3,5}+5H_{3,5}^{2}\Big{)},$

and so on.

${\bf N=3}$: The hierarchy consists of three compatible systems. The first one
is the $N=3$ Novikov equation, the rest are its commuting symmetries:

$\displaystyle\partial_{t_{1}}(u_{s})$ $\displaystyle=\,u_{s+1},\quad
s=0,\ldots,4;$ $\displaystyle\partial_{t_{1}}(u_{5})$
$\displaystyle=\,128\alpha_{8}\\!+\\!35u^{4}\\!+\\!48\alpha_{4}u^{2}\\!-\\!70u_{2}u^{2}\\!-\\!64\alpha_{6}u\\!-\\!16\alpha_{4}u_{2}\\!-\\!70u_{1}^{2}u\\!+\\!14u_{4}u\\!+\\!21u_{2}^{2}\\!+\\!28u_{1}u_{3};$
$\displaystyle 4\partial_{t_{3}}(u_{s})$
$\displaystyle=\,{\mathcal{D}}^{s+1}(u_{2}-3u^{2}),\quad s=0,\ldots,5\,;$
$\displaystyle 16\partial_{t_{5}}(u_{s})$
$\displaystyle=\,{\mathcal{D}}^{s+1}(u_{4}-5u_{1}^{2}-10uu_{2}+10u^{3}+16\alpha_{4}u),\quad
s=0,\ldots,5\,.$

It follows from Proposition 18 that there are three common first integrals of
this systems

$H_{3,7}=\frac{3}{2^{9}}\big{(}14u^{5}+32\alpha_{4}u^{3}-64\alpha_{6}u^{2}-70u_{1}^{2}u^{2}+256\alpha_{8}u-16\alpha_{4}u_{1}^{2}-14u_{2}^{2}u+\\\
+28u_{3}u_{1}u-u_{3}^{2}+28u_{2}u_{1}^{2}+2u_{4}u_{2}-2u_{5}u_{1}\big{)};$
(51)
$H_{5,7}=-\frac{5}{2^{11}}\big{(}70u^{6}+144\alpha_{4}u^{4}-70u_{2}u^{4}-256\alpha_{6}u^{3}-280u_{1}^{2}u^{3}-96\alpha_{4}u_{2}u^{2}+768\alpha_{8}u^{2}-\\\
-14u_{2}^{2}u^{2}+168u_{3}u_{1}u^{2}+128\alpha_{6}u_{2}u+16\alpha_{4}u_{2}^{2}-64\alpha_{6}u_{1}^{2}-256\alpha_{8}u_{2}-20u_{3}^{2}u+\\\
+140u_{2}u_{1}^{2}u+12u_{4}u_{2}u-12u_{5}u_{1}u-35u_{1}^{4}-2u_{2}^{3}-u_{4}^{2}-36u_{3}u_{2}u_{1}+12u_{4}u_{1}^{2}+2u_{5}u_{3}\big{)};$
(52)
$H_{7,7}=\frac{7}{2^{13}}\big{(}300u^{7}+576\alpha_{4}u^{5}-700u_{2}u^{5}-960\alpha_{6}u^{4}-1050u_{1}^{2}u^{4}+70u_{4}u^{4}-960\alpha_{4}u_{2}u^{3}+\\\
+2560\alpha_{8}u^{3}+420u_{2}^{2}u^{3}+560u_{3}u_{1}u^{3}+96\alpha_{4}u_{4}u^{2}+1280\alpha_{6}u_{2}u^{2}-100u_{3}^{2}u^{2}+1400u_{2}u_{1}^{2}u^{2}-\\\
-80u_{4}u_{2}u^{2}-60u_{5}u_{1}u^{2}+256\alpha_{4}u_{2}^{2}u-192\alpha_{4}u_{3}u_{1}u-128\alpha_{6}u_{4}u-2560\alpha_{8}u_{2}u+16\alpha_{4}u_{3}^{2}+\\\
+192\alpha_{4}u_{2}u_{1}^{2}-32\alpha_{4}u_{4}u_{2}-64\alpha_{6}u_{2}^{2}+128\alpha_{6}u_{3}u_{1}-1280\alpha_{8}u_{1}^{2}+256\alpha_{8}u_{4}-20u_{2}^{3}u+4u_{4}^{2}u-\\\
-360u_{3}u_{2}u_{1}u-20u_{4}u_{1}^{2}u+20u_{5}u_{3}u-410u_{2}^{2}u_{1}^{2}-u_{5}^{2}+20u_{3}u_{1}^{3}+2u_{4}u_{2}^{2}-4u_{4}u_{3}u_{1}+40u_{5}u_{2}u_{1}\big{)}.$
(53)

##  2\. KdV hierarchy and Novikov equations on free associative algebra.

###  2.1. KdV hierarchy on free associative algebra.

It is well known that the KdV equation and its hierarchy can be defined on a
free differential algebra ${\mathfrak{B}}_{0}=(\mathbb{C}\langle
u_{0},u_{1},\ldots\rangle,D)$ with infinite number of noncommuting variables
(see for example, [27], [28]). Algebra ${\mathfrak{B}}_{0}$ has monomial
additive basis $\\{u_{\xi}=u_{i_{1}}u_{i_{2}}\ldots
u_{i_{m}}\,|\,i_{k}\in{\mathbb{Z}}_{\geqslant 0},\ m\in{\mathbb{N}}\\}$. It is
graded algebra

${\mathfrak{B}}_{0}={\mathbb{C}}\bigoplus\limits_{n\geqslant
2}{\mathfrak{B}}_{0,n},$ (54)

induced by the grading of the variables $u_{k},\,|u_{k}|=k+2$ for any
$k\geqslant 0$ and therefore $|u_{\xi}|=i_{1}+\cdots+i_{m}+2m=n$. Here
${\mathfrak{B}}_{0,n}$ is a finite dimensional space
${\mathfrak{B}}_{0,n}={\rm Span}_{\mathbb{C}}\langle
u_{\xi}\,;\,|u_{\xi}|=n\rangle$.

The construction of the hierarchy is similar to the commutative case, although
one has to take care on the order of the variables, since $u_{k}\cdot
u_{s}\neq u_{s}\cdot u_{k}$ if $k\neq s$. Starting with the operator
$L=D^{2}-u$, one can find its square root by the formula
${\mathcal{L}}=D+\sum\limits_{n\geqslant 1}I_{1,n}D^{-n}$, where
$I_{1,n}\in{\mathfrak{B}}_{0}$ are non-commutative polynomials. It follows
from the proof of Lemma 2 that formula (24) for the recursive calculation of
the polynomials $I_{1,n}$ is also applicable in the case of a free associative
algebra ${\mathfrak{B}}_{0}$.

Now the initial segment of the series ${\mathcal{L}}$ has the form

${\mathcal{L}}=D-\frac{1}{2}uD^{-1}+\frac{1}{4}u_{1}D^{-2}-\frac{1}{8}(u_{2}+u^{2})D^{-3}+\frac{1}{16}(u_{3}+2u_{1}u+4uu_{1})D^{-4}+\cdots$

Similarly to the commutative case, we introduce fractional powers
${\mathcal{L}}^{2k-1}$ and polynomials $\varrho_{2k}={\rm
res}\,{\mathcal{L}}^{2k-1}$. From the identity
${\mathcal{L}}^{2k+1}=L{\mathcal{L}}^{2k-1}$ follows a formula for the
recursive calculation of the polynomials $\varrho_{2k+2}$. It follows from the
proof of the formulas (28) and (30) that they are applicable in the case of a
free associative algebra ${\mathfrak{B}}_{0}$. However, expressions for
$\varrho_{2k}\in{\mathfrak{B}}_{0}$ are different from expressions for
$\rho_{2k}\in{\mathfrak{A}}_{0},\,k\geqslant 3$,

$\varrho_{2}=-\frac{1}{2}u;\quad\varrho_{4}=\frac{1}{8}(3u^{2}-u_{2});\quad\varrho_{6}=\frac{1}{32}\big{(}5(u_{2}u+uu_{2})+5u_{1}^{2}-10u^{3}-u_{4}\big{)};$
(55)
$\varrho_{8}=\frac{1}{128}\big{(}7(u_{4}u+uu_{4})+14(u_{3}u_{1}+u_{1}u_{3})+21u_{2}^{2}-21(u_{2}u^{2}+u^{2}u_{2})-28uu_{2}u-\\\
-28(u_{1}^{2}u+uu_{1}^{2})-14u_{1}uu_{1}+35u^{4}-u_{6}\big{)};$ (56)

and so on.

######  Definition 19.

The compatible system of equations on the free associative algebra
${\mathfrak{B}}_{0}$

$\partial_{t_{2k-1}}(u)=-2D(\varrho_{2k})$ (57)

is called the KdV hierarchy (similar to the commutative case (32)).

Equations of the KdV hierarchy define the commuting evolutionary derivations
$D_{2k-1}$ of ${\mathfrak{B}}_{0}$. Their action on the variables $u_{n}$ is
given by

$D_{2k-1}(u_{n})=-2D^{n+1}(\varrho_{2k})$

and it can be extended to ${\mathfrak{B}}_{0}$ by the linearity and the
Leibniz rule.

We have

$\partial_{t_{2k-1}}(\varrho_{2n})=\partial_{t_{2n-1}}(\varrho_{2k})={\rm
res}\,[{\mathcal{L}}^{2n-1}_{+},{\mathcal{L}}^{2k-1}_{-}]={\rm
res}\,[{\mathcal{L}}^{2k-1}_{+},{\mathcal{L}}^{2n-1}_{-}].$ (58)

In the non-commutative case ${\rm res}\,[A,B]$ is not any more in the image of
the derivation $D$ and formula (20) should be modified. Let us introduce the
algebra

${\mathfrak{B}}_{0}^{D}=\Big{\\{}A=\sum\limits_{i\leqslant
m}a_{i}D^{i}\,|\,a_{i}\in{\mathfrak{B}}_{0,|a_{m}|+m-i},\,a_{m}\neq
0,\,m\in{\mathbb{Z}}\Big{\\}}=\underset{k\in\mathbb{Z}}{\oplus}{\mathfrak{B}}_{0,k}^{D}\,.$

Let us introduce the homogeneous skew-symmetric bilinear over $\mathbb{C}$
form

$\widehat{\sigma}(\cdot,\cdot)\colon{\mathfrak{B}}_{0}^{D}\otimes{\mathfrak{B}}_{0}^{D}\rightarrow{\mathfrak{B}}_{0},\quad|\widehat{\sigma}(A,B)|=|A|+|B|,$

such that for $n,m\in\mathbb{Z}$

$\widehat{\sigma}(aD^{n},bD^{m})=\begin{cases}\frac{1}{2}\binom{n}{n+m+1}\\!\sum\limits_{s=0}^{n+m}(-1)^{s}(a^{(s)}b^{(n+m-s)}+b^{(n+m-s)}a^{(s)}),\text{if
}n+m\geqslant 0,\,nm<0,\\\\[2.0pt] 0,\qquad\mbox{otherwise}.\end{cases}$ (59)

######  Lemma 20.

For any $A,B\in{\mathfrak{A}}^{D}$, we have $D\big{(}\sigma(A,B)\big{)}={\rm
res}\,[A,B]-\text{$\Delta$}(A,B)$, where

$\Delta(aD^{n},bD^{m})=\begin{cases}\frac{1}{2}\binom{n}{n+m+1}\Big{(}[a,b^{(n+m+1)}]+(-1)^{n+m}[b,a^{(n+m+1)}]\Big{)},\text{
for }n+m\geqslant-1,\\\\[1.0pt] 0,\qquad\mbox{otherwise}.\end{cases}$

###### Proof.

The statement of this lemma is verified by directly calculating the value of
$\Delta(aD^{n},bD^{m})$. ∎

######  Corollary 21.

Let

$A_{2k-1}=\sum_{i=0}^{2k-1}a_{2k-1-i}D^{i},\qquad\mathcal{L}_{-}^{2n-1}=\sum_{j\geqslant
1}I_{2n-1,j}D^{-j},\;n\geqslant 1.$

Then

${\rm
res}[A_{2k-1},\mathcal{L}_{-}^{2n-1}]=D(\widehat{\sigma}_{2k-1,2n-1})-\Delta_{2k,2n-1}$

where

$\widehat{\sigma}_{2k-1,2n-1}=\frac{1}{2}\sum_{i=1}^{2k-1}\sum_{j=1}^{i}\binom{i}{i-j+1}\sum_{s=0}^{i-j}(-1)^{s}\left(a_{2k-1-i}^{(s)}I_{2n-1,j}^{(i-j-s)}+I_{2n-1,j}^{(i-j-s)}a_{2k-1-i}^{(s)}\right),$
(60)
$\Delta_{2k,2n-1}=\frac{1}{2}\sum_{i=1}^{2k-1}\sum_{j=1}^{i}\binom{i}{i-j+1}\left([a_{2k-1-i},I_{2n-1,j}^{(i-j+1)}]+(-1)^{i-j}[a_{2k-1-i}^{(i-j+1)},I_{2n-1,j}]\right).$
(61)

In the non-commutative case the definition of densities of local conservation
laws has to be modified, since

$\partial_{t_{2k+1}}(\varrho_{2n})\in{\rm
Span}[{\mathfrak{B}}_{0},{\mathfrak{B}}_{0}]\oplus D({\mathfrak{B}}_{0}).$

Here ${\rm Span}[{\mathfrak{B}}_{0},{\mathfrak{B}}_{0}]$ is a linear subspace
generated by all commutators of elements from ${\mathfrak{B}}_{0}$.

A $\mathbb{C}$-linear space of functionals is defined as
${\mathfrak{B}}_{0}^{\sharp}={\mathfrak{B}}_{0}/\big{(}{\rm
Span}[{\mathfrak{B}}_{0},{\mathfrak{B}}_{0}]\oplus
D({\mathfrak{B}}_{0})\big{)}$, see [12], [26], [27]. The polynomials
$\varrho_{2n}$ as elements of ${\mathfrak{B}}_{0}^{\sharp}$ are constants of
motion of the nonabelian KdV hierarchy (57).

###  2.2. Cyclic Frobenius algebras over free associative algebra.

Denote by $\xi_{k}=(j_{1},\ldots,j_{k})$ sequences of non-negative integers of
length $k\geqslant 1$. Let $u_{\xi_{k}}=u_{j_{1}}\cdots u_{j_{k}}$. We obtain
$|u_{\xi_{k}}|=2k+\sum\limits_{s=1}^{k}j_{s}$.

We will consider ${\mathfrak{B}}_{0}$ as a graded algebra
${\mathfrak{B}}_{0}=\mathbb{C}\oplus\widetilde{{\mathfrak{B}}}_{0}$, where
$\widetilde{{\mathfrak{B}}}_{0}=\oplus_{m}{\mathfrak{B}}_{0,m},\,m\geqslant
2$, and ${\mathfrak{B}}_{0,m}$ is a graded finite-dimensional
$\mathbb{C}$-linear space with an additive lexicographically by indices
ordered monomial basis $\\{u_{\xi_{k}},\,|u_{\xi_{k}}|=m\\}$.

For example, $\\{u_{3},u_{1}u,uu_{1}\\}$ is a monomial and lexicographically
ordered basis $u_{3}\succ u_{1}u\succ uu_{1}$ in ${\mathfrak{B}}_{0}^{5}$.

Let $\xi_{k}=(j_{1},\ldots,j_{k})$ be a multindex of the monomial
$u_{\xi_{k}}$ and $T_{k}$ be a generator of the cyclic permutation group of
order $k$: $T_{1}(\xi_{1})=\xi_{1}$ and
$T_{k}(\xi_{k})=(j_{2},\ldots,j_{k},j_{1}),\,k\geqslant 2$. Let us denote by
$T(\xi_{k})$ the maximal index set in
$T(\xi_{k})=\max_{\succ}\\{\xi_{k},T_{k}(\xi_{k}),\ldots,T_{k}^{k-1}(\xi_{k})\\}$
with respect to the lexicographic ordering. We define the linear homomorphism
${\mathcal{T}}\colon{\mathfrak{B}}_{0,m}\to{\mathfrak{B}}_{0,m}$ by its action
on the monomial basis elements ${\mathcal{T}}(u_{\xi_{k}})=u_{T(\xi_{k})}$.

For example, ${\mathcal{T}}(u_{1}u)={\mathcal{T}}(uu_{1})=u_{1}u$.

######  Proposition 22.

Homomorphism ${\mathcal{T}}\colon{\mathfrak{B}}_{0}\to{\mathfrak{B}}_{0}$ is a
projector such that $\ker{\mathcal{T}}={\rm
Span}[{\mathfrak{B}}_{0},{\mathfrak{B}}_{0}]$ and
$\rm{Im}{\mathcal{T}}\simeq{\mathfrak{B}}_{0}^{\natural}={\mathfrak{B}}_{0}/{\rm
Span}[{\mathfrak{B}}_{0},{\mathfrak{B}}_{0}]$.

###### Proof.

It follows directly from the definition that
${\mathcal{T}}={\mathcal{T}}^{2}$. The properties of ${\mathcal{T}}$ follow
from the following facts:

  1. (1)

$[u_{\xi_{k}^{\prime}},u_{\xi_{s}^{\prime\prime}}]=u_{\xi_{k.s}}-u_{T^{k}_{k+s}(\xi_{k.s})}$,
where $\xi_{k.s}=(\xi_{k}^{\prime},\xi_{s}^{\prime\prime})$ is the
concatination of $\xi_{k}^{\prime}$ and $\xi_{s}^{\prime\prime}$;

  2. (2)

$u_{\xi_{k}}-u_{T(\xi_{k})}=[u_{j_{1}},u_{j_{2}}\cdots u_{j_{k}}]$ for
$\xi_{k}=(j_{1},\ldots,j_{k}),\,k>1$.

∎

It follows from Proposition 22 that the projector ${\mathcal{T}}$ gives the
splitting of the exact sequence

$0\rightarrow{\rm
Span}[{\mathfrak{B}}_{0},{\mathfrak{B}}_{0}]\rightarrow{\mathfrak{B}}_{0}{\longrightarrow}{\mathfrak{B}}_{0}^{\natural}\rightarrow
0.$ (62)

It enables us to identify the element ${\mathcal{T}}(b),\
b\in{\mathfrak{B}}_{0}$ with its canonical projection in
${\mathfrak{B}}_{0}^{\natural}$.

######  Theorem 23.

The algebra ${\mathfrak{B}}_{0}^{D}$ is the $\rm{CF}$-algebra in which the
bilinear form
$\Phi=\overline{\sigma}\colon{\mathfrak{B}}_{0}^{D}\otimes_{{\mathfrak{B}}_{0}}{\mathfrak{B}}_{0}^{D}\to{\mathfrak{B}}_{0}^{\natural}$
is uniquely given by the formula

$\overline{\sigma}(D^{n},bD^{m})=\begin{cases}\binom{n}{n+m+1}{\mathcal{T}}({b}^{(n+m)}),&\mbox{if
}n+m\geqslant 0,\,nm<0,\\\ 0,&\mbox{otherwise}.\end{cases}$ (63)

###### Proof.

Let us first explain the expression ${\mathcal{T}}({b}^{(n+m)})$. The space
${\rm Span}[{\mathfrak{B}}_{0},{\mathfrak{B}}_{0}]$ is closed under the
differentiation $D$. Therefore, the operator $D$ on ${\mathfrak{B}}_{0}$
uniquely determines the linear operator
$\overline{D}:{\mathfrak{B}}_{0}\mapsto{\mathfrak{B}}_{0}^{\natural}$, and for
any $b\in{\mathfrak{B}}_{0}$ the formula $\overline{D}(b)={\mathcal{T}}(D(b))$
holds.

Let $\xi_{k}=(j_{1},\ldots,j_{k}),\,k\geqslant 1$, and
$T(\xi_{k})=(j_{1,*},\ldots,j_{k,*})$. If
$\xi_{k}=(j_{1},\ldots,j_{1})=(j_{1})^{k},\,k\geqslant 1$, then
$\overline{D}(u_{\xi_{k}})=ku_{j_{1}+1}u_{j_{1}}^{k-1}$. If there are at least
two distinct elements in the set $\xi_{k}$, then $\overline{D}(u_{\xi_{k}})$
is a sum of monomials with the leading monomial
$u_{j_{1}+1,*}u_{j_{2},*}\cdots u_{j_{k,*}}$. Thus, in a strictly ordered
basis, the homomorphism
$\overline{D}\colon{{\mathfrak{B}}}_{0,m}\to{{\mathfrak{B}}}_{0,m+1}^{\natural}$
is given by an upper triangular matrix with a non-zero diagonal. It induces
the monomorphism
$\overline{D}\colon{{\mathfrak{B}}}_{0,m}^{\natural}\to{{\mathfrak{B}}}_{0,m+1}^{\natural}$.
Following the proof of Theorem 9 and using Lemma 20, it is easy to complete
the proof of Theorem 23. ∎

######  Corollary 24.

The algebra ${\mathfrak{B}}_{0}^{D}$ is the $\rm{CF}$-algebra with the
bilinear form $\widehat{\sigma}(A,B)$ (60).

###### Proof.

It follows immediately from Theorem 23 and the fact that
$\widehat{\sigma}(A,B)-\overline{\sigma}(A,B)\in{\rm
Span}[{\mathfrak{B}}_{0},{\mathfrak{B}}_{0}]$ for any
$A,B\in{\mathfrak{B}}_{0}^{D}$. ∎

###  2.3. Non-commutative $N$-th Novikov equation and its hierarchy.

Let ${\mathfrak{B}}=({\mathcal{A}}\langle u,u_{1},\ldots\rangle,D)$ be the
differential ring, where ${\mathcal{A}}={\mathbb{C}}[\alpha_{4},\ldots],\
D(u_{k})=u_{k+1},\ D(\alpha_{2n})=0,$ and $\alpha_{2n}$ are commuting
parameters, i.e centre elements of ${\mathfrak{B}}$. Similar to the
commutative case, we fix a positive integer $N$ and define a homogenous
polynomial ${\mathfrak{F}}_{2N+2}\in{\mathfrak{B}}$:

${\mathfrak{F}}_{2N+2}=\varrho_{2N+2}+\sum_{k=0}^{N-1}\alpha_{2(N-k+1)}\varrho_{2k}.$
(64)

Let ${\mathfrak{I}}_{N}\subset{\mathfrak{B}}$ be a two-sided differential
ideal generated by the polynomials ${\mathfrak{F}}_{2N+2}$ and
$D^{k}({\mathfrak{F}}_{2N+2})$, $k\in\mathbb{N}$. The quotient ring
${\mathfrak{B}}_{N}={\mathfrak{B}}/{{\mathfrak{I}}_{N}}$ is a graded finitely
generated free ring ${\mathfrak{B}}_{N}=\mathcal{A}\langle
u_{0},u_{1},\ldots,u_{2N-1}\rangle$ over ${\mathcal{A}}$. Similar to
Proposition 17 we can show that equations of the hierarchy (57) are all
compatible, therefore $D_{2k-1}({\mathfrak{I}}_{N})\subset{\mathfrak{I}}_{N}$.
The derivations $D_{2k-1}$ induce derivations ${\mathcal{D}}_{{2k-1}}$ of the
quotient algebra ${\mathfrak{B}}_{N}$.

The equation ${\mathfrak{F}}_{2N+2}=0$ can be written in the form

$u_{2N}=\mathcal{G}_{2N+2}(u_{0},u_{1},\ldots,u_{2N-2})=\widehat{\varrho}_{2N+2}+2^{2N+1}\sum_{k=0}^{N-1}\alpha_{2(N-k+1)}\varrho_{2k},$
(65)

which is called the non-commutative $N$-th Novikov equation.

In the non-commutative case the definition of first integrals has to be
modified [22], [28]. Let ${\rm
Span}_{\mathcal{A}}[{\mathfrak{B}}_{N},{\mathfrak{B}}_{N}]$ be a
${\mathcal{A}}$-linear subspace generated by the commutators of all elements
in ${\mathfrak{B}}_{N}$. We would like to emphasise that ${\rm
Span}_{\mathcal{A}}[{\mathfrak{B}}_{N},{\mathfrak{B}}_{N}]$ is not an ideal in
${\mathfrak{B}}_{N}$.

Let ${\mathfrak{B}}^{\natural}_{N}={\mathfrak{B}}_{N}/{\rm
Span}_{\mathcal{A}}[{\mathfrak{B}}_{N},{\mathfrak{B}}_{N}]$ be the
corresponding quotient linear space. It follows from the Leibniz rule that
derivations of ${\mathfrak{B}}_{N}$ are well defined on
${\mathfrak{B}}^{\natural}_{N}$. There is a short split exact sequence
(compare with (62):

$0\ \rightarrow\ {\rm
Span}_{\mathcal{A}}[{\mathfrak{B}}_{N},{\mathfrak{B}}_{N}]\ \rightarrow\
{\mathfrak{B}}_{N}\ \rightarrow{\mathfrak{B}}^{\natural}_{N}\ \rightarrow\ 0.$
(66)

All constructions and results associated with exact sequence (62) carry over
to the case of exact sequence (66).

######  Definition 25.

A non-constant element ${\mathcal{H}}\in{\mathfrak{B}}^{\natural}_{N}$ is
called a first integral of the $N$-th Novikov equation (respectively of the
non-commutative $N$-th Novikov hierarchy) if
${\mathcal{T}}({\mathcal{D}}{\mathcal{H}})=0$ (resp.
${\mathcal{T}}({\mathcal{D}}_{{2k-1}}{\mathcal{H}})=0,\,k=1,2,\ldots,N$).

Thus, an element ${\mathcal{H}}\in{\mathfrak{B}}_{N}$ is a representative of a
first integral, if and only if ${\mathcal{T}}({\mathcal{H}})\neq 0$ and
${\mathcal{T}}({\mathcal{D}}{\mathcal{H}})=0$.

######  Remark 26.

A nonlinear function of first integrals in the non-commutative case may not be
first integral. It is an important difference with the commutative case.

It follows from (58) and Corollary 21 that

$\widehat{{\mathcal{H}}}_{2n+1,2N+1}=\widehat{\sigma}_{2n+1,2N+1}+\sum_{k=1}^{N-1}\alpha_{2N-2k+2}\widehat{\sigma}_{2n+1,2k-1},\quad
n\in{\mathbb{N}}$ (67)

are first integrals of the non-commutative $N$-th Novikov equation. In the
case of free algebra ${\mathfrak{B}}_{N}$ we get infinitely many first
integrals, since they are algebraically independent. Also the KdV hierarchy
(57) reduced to ${\mathfrak{B}}_{N}$ is infinite, since the derivations
${\mathcal{D}}_{2k-1},\,k>N$, cannot be represented as liner combinations of
${\mathcal{D}}_{2k-1},\ 1\leqslant k\leqslant N$, with coefficients from a
ring of constants (as it takes place in the commutative case (47)).

Example. The $N=1$ Novikov equation (Newton equation on the free algebra
${\mathfrak{B}}_{1}=\langle u,u_{1}\rangle$ with the force
$3u^{2}+8\alpha_{4}$)

$\begin{array}[]{llll}\partial_{t_{1}}(u)&=\,u_{1};&\partial_{t_{3}}(u)&=\,0;\\\
\partial_{t_{1}}(u_{1})&=\,3u^{2}+8\alpha_{4};&\partial_{t_{3}}(u_{1})&=\,0;\end{array}$

has an infinite hierarchy of commuting symmetries. The first three non-trivial
symmetry are

$\displaystyle 16\partial_{t_{5}}(u)$
$\displaystyle=\,(u_{1}u^{2}+u^{2}u_{1})-2uu_{1}u-16\alpha_{4}u_{1};$ (68)
$\displaystyle 16\partial_{t_{5}}(u_{1})$
$\displaystyle=\,-(uu_{1}^{2}+u_{1}^{2}u)+2u_{1}uu_{1}-48\alpha_{4}u^{2}-128\alpha_{4}^{2};$
$\displaystyle 32\partial_{t_{7}}(u)$
$\displaystyle=\,8\alpha_{4}uu_{1}+8\alpha_{4}u_{1}u+2u_{1}u^{3}-u_{1}^{3}+2u^{3}u_{1}-u^{2}u_{1}u-uu_{1}u^{2};$
(69) $\displaystyle 32\partial_{t_{7}}(u_{1})$
$\displaystyle=\,128\alpha_{4}^{2}u-8\alpha_{4}u_{1}^{2}+64\alpha_{4}u^{3}-2u^{2}u_{1}^{2}+uu_{1}uu_{1}-2uu_{1}^{2}u+u_{1}u^{2}u_{1}+$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+u_{1}uu_{1}u-2u_{1}^{2}u^{2}+6u^{5};$
$\displaystyle 16\partial_{t_{9}}(u)$
$\displaystyle=\,3\alpha_{4}(u^{2}u_{1}-2uu_{1}u+u_{1}u^{2}-8\alpha_{4}u_{1});$
(70) $\displaystyle 16\partial_{t_{9}}(u_{1})$
$\displaystyle=\,3\alpha_{4}(64\alpha_{4}^{2}+24\alpha_{4}u^{2}+uu_{1}^{2}-2u_{1}uu_{1}+u_{1}^{2}u).$

There are infinitely many algebraically independent first integrals (in the
sense of Definition 25) given by (67). For example, it follows from (67) that
in ${{\mathfrak{B}}}_{1}^{\natural}$

$\displaystyle\widehat{{\mathcal{H}}}_{3,3}$ $\displaystyle=$
$\displaystyle-\frac{3}{16}\left(\frac{1}{2}u_{1}^{2}-u^{3}-8\alpha_{4}u\right);$
(71) $\displaystyle{\mathcal{T}}({\widehat{{\mathcal{H}}}_{5,3}})$
$\displaystyle=$
$\displaystyle{\frac{5}{128}{\mathcal{T}}\left(64\alpha_{4}^{2}+5uu_{1}^{2}-10u_{1}uu_{1}+5u_{1}^{2}u\right)}=\frac{5}{2}\alpha_{4}^{2};$
(72) $\displaystyle{\mathcal{T}}({\widehat{{\mathcal{H}}}_{7,3}})$
$\displaystyle=$
$\displaystyle\frac{7}{265}(u_{1}uu_{1}u-u_{1}^{2}u^{2})-\frac{7}{3}\alpha_{4}\widehat{{\mathcal{H}}}_{3,3};$
(73) $\displaystyle{\mathcal{T}}({\widehat{{\mathcal{H}}}_{9,3}})$
$\displaystyle=$
$\displaystyle\frac{9}{512}\left(u_{1}uu_{1}u^{2}-u_{1}^{2}u^{3}\right)-\frac{9}{2}\alpha_{4}^{3}+\frac{1}{2}{\mathcal{T}}({\widehat{{\mathcal{H}}}_{3,3}^{2}}).$
(74)

We have

$\displaystyle{\mathcal{D}}(\widehat{{\mathcal{H}}}_{3,3})=-\frac{3}{64}(u^{2}u_{1}-2uu_{1}u+u_{1}u^{2});\qquad\
{\mathcal{D}}({\mathcal{T}}({\widehat{{\mathcal{H}}}_{5,3}}))=0;$
$\displaystyle{\mathcal{D}}({\mathcal{T}}({\widehat{{\mathcal{H}}}_{7,3}}))=\frac{7}{256}\left(8\alpha_{4}uu_{1}u-8\alpha_{4}u_{1}u^{2}+u_{1}uu_{1}^{2}-u_{1}^{2}uu_{1}+3u^{3}u_{1}u-3u^{2}u_{1}u^{2}\right)-\frac{7}{3}\alpha_{4}{\mathcal{D}}(\widehat{{\mathcal{H}}}_{3,3}),$

and
${\mathcal{T}}({{\mathcal{D}}(\widehat{{\mathcal{H}}}_{3,3})})={\mathcal{T}}({{\mathcal{D}}(\widehat{{\mathcal{H}}}_{5,3})})={\mathcal{T}}({{\mathcal{D}}(\widehat{{\mathcal{H}}}_{7,3})})={\mathcal{T}}({{\mathcal{D}}(\widehat{{\mathcal{H}}}_{9,3})})=0$.

Note, that $(\widehat{{\mathcal{H}}}_{3,3})^{2}$ is not a first integral of
the $N=1$ Novikov hierarchy (Remark 26).

###  2.4. Self-adjointness of the KdV and $N$–Novikov hierarchies.

Let ${\mathfrak{B}}_{\mathcal{A}}={\mathcal{A}}\langle u,u_{1},\ldots\rangle$.
The set of differential operators

${\mathfrak{B}}_{\mathcal{A}}[D]=\left\\{A_{+}=\sum_{i=0}^{m}a_{i}D^{i}\,|\,a_{i}\in{\mathfrak{B}}_{\mathcal{A}},\,a_{m}\neq
0,\,m\in\mathbb{Z}_{\geqslant 0}\right\\}$

and the set of differential formal series

${\mathfrak{B}}_{\mathcal{A}}^{D}={\mathfrak{B}}_{\mathcal{A}}[D][[D^{-1}]]=\left\\{A=\sum_{i\leqslant
m}a_{i}D^{i}\,|\,a_{i}\in{\mathfrak{B}}_{\mathcal{A}},\,a_{m}\neq
0,\,m\in\mathbb{Z}\right\\}$

are non-commutative associative algebras in which multiplication is defined by
formula (14). According to formula (14), a conjugation anti-automorphism

${\dagger}\colon{\mathfrak{B}}_{\mathcal{A}}^{D}\to{\mathfrak{B}}_{\mathcal{A}}^{D}\,:\,(AB)^{\dagger}=B^{\dagger}A^{\dagger}$

is defined on the ring ${\mathfrak{B}}_{\mathcal{A}}^{D}$.

######  Lemma 27.

  1. 1)

The operator ${\dagger}$ is uniquely defined by the conditions
$u^{\dagger}=u,\,D^{\dagger}=-D,\,\alpha_{2k}^{\dagger}=\alpha_{2k},\,z^{\dagger}=\bar{z},\,z\in\mathbb{C}$.

  2. 2)

On the ring ${\mathfrak{B}}_{\mathcal{A}}$ the operators $D$ and ${\dagger}$
commute, i.e.

$(D(a))^{\dagger}=D(a^{\dagger})\;\text{ for any
}a\in{\mathfrak{B}}_{\mathcal{A}}.$ (75)

###### Proof.

We have $u_{1}=Du-uD$. Therefore, $u_{1}^{\dagger}=-uD+Du=u_{1}$. Using the
formula $u_{k+1}=Du_{k}-u_{k}D$, by induction on $k,\,k\geqslant 1$, we obtain
that $u_{k+1}^{\dagger}=u_{k+1}$. Using now the formula $DD^{k}=D^{k+1}$, by
induction on $k\geqslant 1$ and $k\leqslant-1$ we obtain that
$(D^{k})^{\dagger}=(-1)^{k}D^{k}$, where $k\in\mathbb{Z}$ and $D^{0}=1$ is the
identity operator.

Assertion 1) now follows from the fact that elements $u_{k},\,k\geqslant 0$,
$D$ and $D^{-1}$, where $DD^{-1}=1$, multiplicatively generate the ring
${\mathfrak{B}}_{\mathcal{A}}^{D}$ as a module over the ring ${\mathcal{A}}$.

Assertion 2) is verified directly. Let $a\in{\mathfrak{B}}_{\mathcal{A}}$.
Then

$(D(a))^{\dagger}=(Da-
aD)^{\dagger}=-a^{\dagger}D+Da^{\dagger}=D(a^{\dagger}).$

∎

A differential series $A\in{\mathfrak{B}}_{\mathcal{A}}^{D}$ is called self-
adjoint if $A^{\dagger}=A$, and anti-self-adjoint if $A^{\dagger}=-A$.

Examples.

1. The operator $L=D^{2}-u$ is self-adjoint.

2. The series $\mathcal{L}=D-\frac{1}{2}uD^{-1}+\ldots,\,\mathcal{L}^{2}=L$, is anti-self-adjoint.

Let $A,B\in{\mathfrak{B}}_{\mathcal{A}}^{D}$. Then

$[A,B]^{\dagger}=(AB)^{\dagger}-(BA)^{\dagger}=B^{\dagger}A^{\dagger}-A^{\dagger}B^{\dagger}=-[A^{\dagger},B^{\dagger}].$

3. Let $A$ and $B$ be self-adjoint (or anti-self-adjoint) series. Then series $[A,B]$ is anti-self-adjoint.

4. Let one of the series $A$ or $B$ be self-adjoint and the other be anti-self-adjoint. Then series $[A,B]$ is self-adjoint.

Consider the series $A=\sum\limits_{i\leqslant m}a_{i}D^{i}$. From the formula
(14), we obtain

$A^{\dagger}=\sum_{i\leqslant m}(-1)^{i}D^{i}a_{i}^{\dagger}=\sum_{i\leqslant
m}(-1)^{i}\sum_{j\geqslant 0}\binom{i}{j}D^{j}(a_{i}^{\dagger})D^{i-j}.$ (76)

According to (76), from conditions $j\geqslant 0$ and $i=j-1$ we obtain that
$j=0$ and $i=-1$, i.e.

${\rm res}(A^{\dagger})=-({\rm res}A)^{\dagger}.$ (77)

The representation $A=A_{+}+A_{-}$ is uniquely defined. So according to (76),
we have

$(A_{+})^{\dagger}=(A^{\dagger})_{+}\ \,\;\text{ and
}\;(A_{-})^{\dagger}=(A^{\dagger})_{-}$ (78)

Explicit formulas $\varrho_{2k}^{\dagger}=\varrho_{2k}$ in the case
$k=1,2,3,4$ see (55) and (56).

######  Lemma 28.

The operators $\partial_{t_{2k-1}},\,k=1,2,\ldots$, participating in the
noncommutative hierarchy KdV, commute with the conjugation operator †, i.e.

$\big{(}\partial_{t_{2k-1}}(a)\big{)}^{\dagger}=\partial_{t_{2k-1}}(a^{\dagger})$
(79)

for any $a\in{\mathfrak{B}}_{\mathcal{A}}$.

###### Proof.

Applying the Leibniz rule for the operator $\partial_{t_{2k-1}}$, we find that
it suffices to prove formula (79) only for multiplicative generators
$u_{i},\,i\geqslant 0$, of the ring ${\mathfrak{B}}_{\mathcal{A}}$. Since
$u_{i}=D^{i}(u)$ and the operators $\partial_{t_{2k-1}}$ commute with the
operator $D$, it suffices to verify formula (79) only for the case $a=u$. By
the definition of the hierarchy KdV, on a free associative algebra we have
$\partial_{t_{2k-1}}(u)=-2D(\varrho_{2k})$ where by definition
$\varrho_{2k}={\rm res}\mathcal{L}^{2k-1}$. Using the formula (75) and
Corollary 20, we obtain

$\big{(}\partial_{t_{2k-1}}(u)\big{)}^{\dagger}=-2\big{(}D(\varrho_{2k})\big{)}^{\dagger}=-2D(\varrho_{2k}^{\dagger})=-2D(\varrho_{2k})=\partial_{t_{2k-1}}(u).$

∎

Let us now turn to the case of the $N$–Novikov hierarchy described in Section
2.3.

######  Lemma 29.

  1. 1)

For any $N\geqslant 1$, the conjugation operator
${\dagger}\colon{\mathfrak{B}}_{N}\to{\mathfrak{B}}_{N}$ is defined.

  2. 2)

The operators $D_{2k-1}$, of the $N$–Novikov hierarchy, commute with the
operator ${\dagger}$.

###### Proof.

We have ${\mathfrak{B}}_{N}={\mathfrak{B}}/{\mathfrak{I}}_{N}$ where
${\mathfrak{I}}_{N}$ is a two-sided ideal generated by polynomials
${\mathfrak{F}}_{2N+2}$ and $D^{k}({\mathfrak{F}}_{2N+2}),\,k\in\mathbb{N}$.
Since $\varrho_{2k}^{\dagger}=\varrho_{2k}$ and
$\big{(}D(a)\big{)}^{\dagger}=D(a^{\dagger})$, then all generators of the
ideal ${\mathfrak{I}}_{N}$ are self-adjoint polynomials. This proves assertion
1).

Assertion 2) follows directly from Lemma 28. ∎

Let us consider the short exact sequence (66). The operator
${\dagger}\colon{\mathfrak{B}}_{N}\to{\mathfrak{B}}_{N}$ moves the linear
space ${\rm Span}_{\mathcal{A}}[{\mathfrak{B}}_{N},{\mathfrak{B}}_{N}]$ into
itself. Therefore, the conjugation operator
${\dagger}\colon{\mathfrak{B}}^{\natural}_{N}\to{\mathfrak{B}}^{\natural}_{N}$
is defined. The first integrals of the $N$–Novikov hierarchy are given by
formula (67), where by definition
$\widehat{\sigma}_{2n+1,2k-1}=\widehat{\sigma}(\mathcal{L}_{+}^{2n+1},\mathcal{L}_{-}^{2k-1})$.

######  Proposition 30.

For $k,n\in{\mathbb{N}}$

  1. 1)

$\varrho_{2k}=\varrho_{2k}^{\dagger}$.

  2. 2)

$\widehat{\sigma}_{2n+1,2k-1}=\widehat{\sigma}_{2n+1,2k-1}^{\dagger}$.

###### Proof.

The property 1): $\varrho_{2k}^{\dagger}=\big{(}{\rm
res}\mathcal{L}^{2k-1}\big{)}^{\dagger}=-{\rm
res}\big{(}(\mathcal{L}^{2k-1})^{\dagger}\big{)}={\rm
res}\mathcal{L}^{2k-1}=\varrho_{2k}$. The property 2) follows from 1) and
Lemma 28. ∎

######  Proposition 31.

The self-adjoint polynomials

$\widehat{{\mathcal{H}}}_{2n+1,2N+1}=\sum_{k=1}^{N+1}\alpha_{2N-2k+2}\,\widehat{\sigma}_{2n+1,2k-1},\;\text{
where }\alpha_{0}=1,$ (80)

are first integrals of the $N$–Novikov hierarchy.

The proof follows from (58), Corollary 21, Lemma 29 and Proposition 30.

##  3\. Quantisation of Novikov equations.

###  3.1. Quantum Novikov equations.

In this section we define a quantisation ideal ${\mathfrak{Q}}_{N}$ and the
quantum $N$-th Novikov equation. Let ${\mathcal{Q}}_{N}$ be a commutative
graded algebra of parameters

${\mathcal{Q}}_{N}={\mathbb{C}}[\alpha_{2j+2},q_{i,j},q_{i,j}^{\omega}\,|\;0\leqslant
i<j\leqslant 2N-1,\;0\leqslant|\omega|<i+j+4]$

where $|q_{i,j}|=0$,
$\omega=(i_{2N-1},\ldots,i_{1},i_{0})\in{\mathbb{Z}}_{\geqslant}^{2N}$,
$|\omega|=(2N+1)i_{2N-1}+\cdots+3i_{1}+2i_{0}$,
$|q_{i,j}^{\omega}|=i+j+4-|\omega|$. The parameters are constant in a sense
that for any $a\in{\mathcal{Q}}_{N}$ we have ${\mathcal{D}}_{2k-1}(a)=0$. Let
$u^{\omega}=u_{2N-1}^{i_{2N-1}}\cdots u_{1}^{i_{1}}u_{0}^{i_{0}}$, and thus
$|u^{\omega}|=|\omega|$. Let ${\mathfrak{B}}_{N}(q)$ denote the graded ring

${\mathfrak{B}}_{N}(q)={\mathcal{Q}}_{N}\langle u_{0},\ldots,u_{2N-1}\rangle.$

with parameters, which are in the centre of the ring.

Let ${\mathfrak{Q}}_{N}=\langle p_{i,j}\,|\,0\leqslant i<j\leqslant
2N-1\rangle\subset{\mathfrak{B}}_{N}(q)$ be a two-sided
${\mathcal{D}}={\mathcal{D}}_{1}$ differential homogeneous ideal generated by
the polynomials

$p_{i,j}=u_{i}u_{j}-q_{i,j}u_{j}u_{i}+\sum\limits_{0\leqslant|\omega|<i+j+4}q_{i,j}^{\omega}u^{\omega},\quad
0\leqslant i<j\leqslant 2N-1,\quad q_{i,j}\neq 0.$ (81)

Let us consider the graded associative algebra
${\mathfrak{C}}_{N}={\mathfrak{B}}_{N}(q)/{\mathfrak{Q}}_{N}$ and the ring
epimorphism ${\mathfrak{B}}_{N}(q)\to{\mathfrak{C}}_{N}$ preserving the
grading.

######  Definition 32.

The ideal ${\mathfrak{Q}}_{N}$ is called the Poincaré–Birkhoff–Witt ideal
(briefly, the PBW-ideal) if the image of the set of monomials $u^{\omega},\
\omega\in{\mathbb{Z}}_{\geqslant}^{2N}$, forms a non-degenerate additive basis
in the ${\mathcal{Q}}$-module ${\mathfrak{C}}_{N}$.

######  Definition 33.

The ideal ${\mathfrak{Q}}_{N}$ is a _quantisation ideal_ of the $N$-th Novikov
equation if:

  1. 1)

it is the PBW-ideal;

  2. 2)

it is invariant with respect to the derivation ${\mathcal{D}}$.

Condition 2) of Definition 33 reduces to a system of polynomial algebraic
equations in ${\mathcal{Q}}_{N}$.

######  Lemma 34.

Let ${\mathfrak{Q}}_{N}\subset{\mathfrak{B}}_{N}$ be a quantisation ideal of
the $N$-th Novikov equation. Then $q_{i,j}=1$.

###### Proof.

The Lemma can be proven by induction. Let us show that $q_{2N-2,2N-1}=1$.
Applying ${\mathcal{D}}$ to the polynomial $p_{2N-2,2N-1}$ we get

${\mathcal{D}}(p_{2N-2,2N-1})=(1-q_{2N-2,2N-1})u_{2N-1}^{2}+f_{2N-2,2N-1}.$

where $f_{2N-2,2N-1}$ is a polynomial whose leading monomial ${\rm
Lm}(f_{2N-2,2N-1})<u_{2N-1}^{2}$. Thus $q_{2N-2,2N-1}=1$ is the necessary
condition for ${\mathcal{D}}({\mathfrak{Q}}_{N})\subset{\mathfrak{Q}}_{N}$.
Let us assume that $q_{i,j}=1$ for all $i<j$, such that $i+j\geqslant k$. For
a polynomial $p_{i,j}$ with $i<j$ and $i+j<k$ we get

${\mathcal{D}}(p_{i,j})=(u_{i+1}u_{j}+u_{i}u_{j+1})-q_{i,j}(u_{j+1}u_{i}+u_{j}u_{i+1})+\sum
q_{i,j}^{\omega}{\mathcal{D}}(u^{\omega}).$

By the induction assumption $q_{i,j+1}=1$. Therefore,
${\mathcal{D}}(p_{i,j})=(1-q_{i,j})u_{j+1}u_{i}+f_{i,j}$, where $f_{i,j}$ is a
polynomial such that ${\rm Lm}(f_{i,j})\prec u_{j+1}u_{i}$ in the additive
basis of ${\mathfrak{C}}_{N}$. It follows from
${\mathcal{D}}(p_{i,j})\in{\mathfrak{Q}}_{N}$ that $q_{i,j}=1$. ∎

######  Corollary 35.

  1. 1)

The relation

$[u_{i},u_{j}]=-h_{ij},\quad 0\leqslant i<j\leqslant 2N-1,$

holds in the ring ${\mathfrak{C}}_{N}$, where

$h_{ij}=\sum_{0\leqslant|\omega|<i+j+4}q_{i,j}^{\omega}u^{\omega},\quad
q_{i,j}^{\omega}\in{\mathcal{Q}}_{N}.$

  2. 2)

For any $P\in{\mathfrak{C}}_{N}$, polynomial $[u_{k},P]\in{\mathfrak{C}}_{N}$
is a linear combination of polynomials $h_{ij}$ with coefficients from
${\mathfrak{C}}_{N}$.

Since the ideal ${\mathfrak{Q}}_{N}\subset{\mathfrak{B}}_{N}(q)$ is
${\mathcal{D}}$-invariant, i.e.
${\mathcal{D}}({\mathfrak{Q}}_{N})\subset{\mathfrak{Q}}_{N}$, then the
derivation ${\mathcal{D}}$ induces a well defined derivation
$\partial_{t_{1}}$ on the quotient algebra
${\mathfrak{B}}_{N}(q)/{\mathfrak{Q}}_{N}$ and a quantum dynamical system
defined by the quantum $N$-th Novikov equation

$\partial_{t_{1}}(u_{k})=u_{k+1},\;k=0,\ldots,2N-2,\qquad\partial_{t_{1}}(u_{2N-1})={\mathfrak{G}}_{2N+2}(u_{0},\ldots,u_{2N-2}),$

where
${\mathfrak{G}}_{2N+2}(u_{0},\ldots,u_{2N-2})\in{\mathfrak{B}}_{N}(q)/{\mathfrak{Q}}_{N}$
is a canonical projection of
${\mathcal{G}}_{2N+2}(u_{0},\ldots,u_{2N-2})\in{\mathfrak{B}}_{N}$.

######  Theorem 36.

The ideal ${\mathfrak{Q}}_{N}$ is the quantisation ideal of the $N$-th Novikov
equation if and only if the set of polynomials $h_{ij}\in{\mathfrak{C}}_{N}$
is a solution of the following systems in the ring ${\mathfrak{C}}_{N}$:

  1. I.

the system of algebraic equations linear in $h_{ij}$

$[h_{ij},u_{k}]+[h_{jk},u_{i}]=[h_{ik},u_{j}]$ (82)

for all triples $(i,j,k),\;0\leqslant i<j<k\leqslant 2N-1$;

  2. II.

the system of differential equations linear in $h_{ij}$

$h_{ij}^{\prime}=\widetilde{h}_{i+1,j}+\widetilde{h}_{i,j+1}$ (83)

where $h_{ij}^{\prime}=\partial_{t_{1}}(h_{ij})$ and

$\widetilde{h}_{i+1,j}=\begin{cases}h_{i+1,j},&\mbox{if }\,i+1<j;\\\
0,&\mbox{if
}\,i+1=j;\end{cases}\qquad\widetilde{h}_{i,j+1}=\begin{cases}h_{i,j+1},&\mbox{if
}\,j+1<2N;\\\ [u_{i},{\mathfrak{G}}_{2N+2}],&\mbox{if }\,j=2N-1.\end{cases}$

###### Proof.

In [20], V. Levandovskyy obtained necessary and sufficient conditions on
polynomials $p_{i,j}$ of the form (81) under which the ideal
${\mathfrak{Q}}_{N}$ is the BPW-ideal. Under the additional condition
$q_{i,j}=1$ (see Lemma 34), Lemma 2.1 from [20] provides a proof of Statement
I. The proof of Statement II follows from Statements 1)–2) of Corollary 35. ∎

Let the ideal ${\mathfrak{Q}}_{N}\subset{\mathfrak{B}}_{N}$ be a quantization
ideal that is invariant under derivations
${\mathcal{D}}_{2k-1},\,k=2,\ldots,N$, on the ring ${\mathfrak{B}}_{N}$. Then
derivations $\partial_{t_{2k-1}},\,k=2,\ldots,N$, are defined on the ring
${\mathfrak{C}}_{N}$ such that
$\partial_{t_{2k-1}}(u)=\partial_{t_{1}}(\widehat{\varrho}_{2k})$, where
$\widehat{\varrho}_{2k}\in{\mathfrak{C}}_{N}$ is the image of the polynomial
$\varrho_{2k}\in{\mathfrak{B}}_{N}$.

######  Corollary 37.

The polynomials $h_{ij}\in{\mathfrak{C}}_{N}$ satisfy the following system of
differential equations linear in $h_{ij}$

$\partial_{t_{2k-1}}(h_{ij})=[u_{i},\partial_{t_{1}}^{j+1}(\widehat{\varrho}_{2k})]-[u_{j},\partial_{t_{1}}^{i+1}(\widehat{\varrho}_{2k})],\quad
k=1,\ldots,N,\;0\leqslant i<j\leqslant 2N-1.$

###  3.2. Quantum $N=1$ Novikov equation.

Let

$u_{2}={\mathcal{G}}_{4}(u_{0})=3u^{2}+8\alpha_{4}$ (84)

and ${\mathfrak{I}}_{1}\subset{\mathfrak{B}}$ is the two-sided differential
ideal generated by Novikov equation relatively the derivation $D$, such that
$D(u_{k})=u_{k+1}$. Then in
${\mathfrak{B}}_{1}={\mathfrak{B}}/{\mathfrak{I}}_{1}=\mathcal{A}\langle
u_{0},u_{1}\rangle$ the induced derivation ${\mathcal{D}}$ can be defined by
its action on the generators

${\mathcal{D}}(u)=u_{1},\quad{\mathcal{D}}(u_{1})=3u^{2}+8\alpha_{4}.$

Let us consider a homogeneous two sided ideal
${\mathfrak{Q}}_{1}\subset{\mathfrak{B}}_{1}(q)$ generated by one polynomial

$p_{0,1}=uu_{1}-u_{1}u-q^{(0,2)}u^{2}-q^{(1,0)}u_{1}-q^{(0,1)}u-q^{(0,0)}$

with arbitrary constants $q^{\omega}$. Let us find conditions on these
constants under which the ideal ${\mathfrak{Q}}_{1}$ becomes the quantization
ideal. According to Theorem 36 we obtain

$h_{01}^{\prime}=[u,3u^{2}+8\alpha_{4}]=0.$

Thus, $q^{(0,2)}=q^{(1,0)}=q^{(0,1)}=0$ and $q^{(0,0)}$ is a free parameter.
Let us denote $q^{(0,0)}=8i\hbar$, then the quotient algebra
${\mathfrak{C}}_{1}={\mathfrak{B}}_{1}(q)/{\mathfrak{Q}}_{1}$ coincides with
the Heisenberg (Weyl) algebra ${\mathbb{C}}[\alpha_{4},\hbar]\langle
u,u_{1}\rangle/\langle uu_{1}-u_{1}u-8i\hbar\rangle$ in quantum mechanics.
Thus we have proved the following statement.

######  Proposition 38.

The ideal ${\mathfrak{Q}}_{1}\subset{\mathfrak{B}}_{1}(q)$ is the quantization
ideal if and only if $uu_{1}-u_{1}u=8i\hbar$, where $\hbar$ is an arbitrary
parameter.

In the case $N=1$ the $N$-th Novikov equation $u_{2}=3u^{2}+8\alpha_{4}$ has
the form of the classical Newton equation. According to Proposition 38, the
quantum $N=1$ Novikov equation is unique and can be written in the Heisenberg
form

$i\hbar\partial_{t_{1}}(u)=[u,{\mathfrak{H}}_{3,3}]=i\hbar u_{1},\qquad
i\hbar\partial_{t_{1}}(u_{1})=[u,{\mathfrak{H}}_{3,3}]=i\hbar(3u^{2}+8\alpha_{4}),$
(85)

where the Hamiltonian operator
${\mathfrak{H}}_{3,3}=-\frac{2}{3}\widehat{{\mathcal{H}}}_{3,3}=\frac{1}{16}(u_{1}^{2}-2u^{3}-16\alpha_{4}u)$
is self-adjoint and $[u,u_{1}]=8i\hbar$. Here $\widehat{{\mathcal{H}}}_{3,3}$
is given by (80). It follows from (85) that the $t_{1}$ derivative of any
element of $a\in{\mathfrak{C}}_{1}$ can be written in the form
$\partial_{t_{1}}(u)=\frac{i}{\hbar}[{\mathfrak{H}}_{3,3},a]$. In particular
the Hamiltonian ${\mathfrak{H}}_{3,3}$ is a quantum constant of motion
$\partial_{t_{1}}({\mathfrak{H}}_{3,3})=\frac{i}{\hbar}[{\mathfrak{H}}_{3,3},{\mathfrak{H}}_{3,3}]=0$.

We have $\partial_{t_{3}}(u)=0$. Higher symmetries (68), (69),(70) on the free
associative algebra ${\mathfrak{B}}_{1}$ after the reduction to
${\mathfrak{C}}_{1}$ take the self-adjoint form

$\displaystyle\partial_{t_{5}}(u)$ $\displaystyle=\,-\alpha_{4}u_{1};$
$\displaystyle\partial_{t_{5}}(u_{1})$
$\displaystyle=\,-\alpha_{4}(3u^{2}+8\alpha_{4});$ $\displaystyle
4\partial_{t_{7}}(u)$
$\displaystyle=\,{\mathfrak{H}}_{3,3}\,u_{1}+u_{1}\,{\mathfrak{H}}_{3,3};$
$\displaystyle 4\partial_{t_{7}}(u_{1})$
$\displaystyle=\,{\mathfrak{H}}_{3,3}\,(3u^{2}+8\alpha_{4})+(3u^{2}+8\alpha_{4})\,{\mathfrak{H}}_{3,3};$
$\displaystyle 2\partial_{t_{9}}(u)$ $\displaystyle=\,3\alpha_{4}^{2}u_{1};$
$\displaystyle 2\partial_{t_{9}}(u_{1})$
$\displaystyle=\,3\alpha_{4}^{2}(3u^{2}+8\alpha_{4}).$

###  3.3. Quantum $N=2$ Novikov hierarchy.

In the case $N=2$ the $N$-th Novikov equation on the free associative algebra
${\mathfrak{B}}={\mathbb{C}}[\alpha_{4},\alpha_{6}]\langle
u,u_{1},\ldots\rangle$ can be written in the form

$u_{4}=\mathcal{G}_{6}(u_{0},u_{1},u_{2}),$ (86)

where

$\mathcal{G}_{6}=5(u_{2}u+uu_{2})+5u_{1}^{2}-10u^{3}-16\alpha_{4}u+32\alpha_{6}.$
(87)

It defines two commuting derivations ${\mathcal{D}},{\mathcal{D}}_{3}$ on the
quotient ring ${\mathfrak{B}}_{2}={\mathbb{C}}[\alpha_{4},\alpha_{6}]\langle
u,u_{1},u_{2},u_{3}\rangle={\mathfrak{B}}/{\mathfrak{I}}_{2}$ which results in
the $2$-KdV hierarchy consisting of two compatible nonabelian systems. The
first system of the hierarchy ($\partial_{t_{1}}(u_{k})={\mathcal{D}}(u_{k})$)

$\partial_{t_{1}}(u)=u_{1};\qquad\partial_{t_{1}}(u_{1})=u_{2};\qquad\partial_{t_{1}}(u_{2})=u_{3};\qquad\partial_{t_{1}}(u_{3})=\mathcal{G}_{6}$
(88)

is equation (86) written in the form of a first order system. The second
system of the hierarchy is

$4\partial_{t_{3}}(u_{k})=\partial_{t_{1}}^{k+1}(u_{2}-3u^{2}),\quad
k=0,1,2,3.$ (89)

Let us introduce a two-sided ideal

$\widehat{{\mathfrak{Q}}}_{2}=\langle[u_{i},u_{j}]-h_{ij};\,[h_{ij},u_{k}];\,0\leqslant
i<j\leqslant 3,\,0\leqslant k\leqslant 3\rangle\subset{\mathfrak{B}}_{2}(q)$

and set

$\widehat{{\mathfrak{C}}}_{2}={\mathfrak{B}}_{2}(q)/\widehat{{\mathfrak{Q}}}_{2}.$

######  Lemma 39.

The ideal $\widehat{{\mathfrak{Q}}}_{2}$ is a quantisation ideal of the $2$-nd
Novikov equation if and only if polynomials
$h_{ij}\in\widehat{{\mathfrak{C}}}_{2}$ are the solution of the following
linear in $h_{ij}$ system of differential equations:

$\displaystyle
h_{01}^{\prime}=h_{02};\qquad\quad\,h_{02}^{\prime}=h_{12}+h_{03};\quad
h_{12}^{\prime}=h_{13};$ $\displaystyle h_{03}^{\prime}=h_{13}+P_{8};\quad
h_{13}^{\prime}=h_{23}+P_{9};\quad\;h_{23}^{\prime}=P_{10},$

where $h_{ij}^{\prime}=\partial_{t_{1}}(h_{ij})$,
$P_{8}=(h_{01}u)^{\prime}=h_{02}u+h_{01}u_{1}$,
$P_{9}=10(h_{12}u-h_{01}u_{2})+16\alpha_{4}h_{01}$,
$P_{10}=10h_{02}(-u_{2}+3u^{2})-10h_{12}u_{1}+16\alpha_{4}h_{02}$.

###### Proof.

Using the formula

$\mathcal{G}_{6}=10(u_{2}u-u^{3})+5(h_{02}+u_{1}^{2})-16\alpha_{4}u+32\alpha_{6}\in\widehat{{\mathfrak{C}}}_{2},$

we obtain that the assertion of this lemma is an immediate consequence of
Theorem 36. ∎

######  Proposition 40.

The ideal ${\mathfrak{Q}}_{2}\subset{\mathfrak{B}}_{2}(q)$ is the quantization
ideal if and only if

$[u_{i},u_{j}]=0\,\text{ for }i+j<3\,\text{ or
}i+j=4;\quad[u,u_{3}]=[u_{2},u_{1}]=32i\hbar;\quad[u_{2},u_{3}]=320i\hbar u,$

where $\hbar$ is an arbitrary parameter.

###### Proof.

Set $h_{01}=\xi\in\widehat{{\mathfrak{C}}}_{2}$. Then, accordingly to Lemma
39, in the ring $\widehat{{\mathfrak{C}}}_{2}$ we have:

$\displaystyle h_{02}$ $\displaystyle=\xi^{\prime};\quad
2h_{03}=\xi^{\prime\prime}+10\xi u+2\alpha,\,\text{ where
}\,\alpha=const,\,|\alpha|=7;$ $\displaystyle 2h_{12}$
$\displaystyle=\xi^{\prime\prime}-10\xi u-2\alpha;\quad
2h_{13}=\xi^{\prime\prime\prime}-10(\xi u)^{\prime};$ $\displaystyle 2h_{23}$
$\displaystyle=\xi^{(4)}-10(\xi u)^{\prime\prime}-20(h_{12}u-\xi
u_{2})-16\alpha_{4}\xi;$ $\displaystyle\xi^{(5)}$ $\displaystyle-10(\xi
u)^{\prime\prime\prime}-20(h_{12}u-\xi
u_{2})^{\prime}-16\alpha_{4}\xi^{\prime}=20\xi^{\prime}(-u_{2}+3u^{2})-20h_{12}u_{1}+32\alpha_{4}\xi^{\prime}.$

Therefore, the polynomial $\xi$ must satisfy the equation

$\xi^{(5)}-20\xi^{\prime\prime\prime}u-30\xi^{\prime\prime}u_{1}+\xi^{\prime}(10u_{2}+40u^{2}-48\alpha_{4})+10\xi(u_{3}+10u_{1}u)=0.$
(90)

Accordingly to formula (81), the solution to this equation should be sought in
the form:

$\xi=\beta_{1}u^{2}+\beta_{2}u_{1}+\beta_{3}u+\beta_{5},\quad|\beta_{k}|=k.$
(91)

Substituting expression (91) into equation (90) and using the PBW-basis in
$\widehat{{\mathfrak{C}}}_{2}$, we obtain that $\xi=0$ in
$\widehat{{\mathfrak{C}}}_{2}$. Then the polynomial
$\xi\in{\mathfrak{B}}_{2}(q)$ must belong to the ideal
$\widehat{{\mathfrak{Q}}}_{2}$, but this is possible only when $[u,u_{1}]=0$.

Under condition $[u,u_{1}]=0$, the system of differential equations (83) in
the case $N=2$ takes the form

$\displaystyle h_{01}$ $\displaystyle=0;\quad h_{02}=0;\quad h_{12}+h_{03}=0;$
$\displaystyle h_{13}$ $\displaystyle=h_{12}^{\prime}=h_{03}^{\prime};\quad
h_{23}=h_{13}^{\prime}-5(h_{12}u+uh_{12});$ $\displaystyle h_{23}^{\prime}$
$\displaystyle=-5(h_{12}u_{1}+u_{1}h_{12}).$

If $\alpha=-32i\hbar$, then we obtain:

$[u,u_{3}]=[u_{2},u_{1}]=32i\hbar;\quad[u_{2},u_{3}]=320i\hbar u.$

∎

######  Proposition 41.

The following statements are equivalent
1. The ideal ${\mathfrak{Q}}_{2}$ is ${\mathcal{D}}$-invariant: ${\mathcal{D}}({\mathfrak{Q}}_{2})\subseteq{\mathfrak{Q}}_{2}$.
2. The ideal ${\mathfrak{Q}}_{2}$ is ${\mathcal{D}}_{3}$-invariant: ${\mathcal{D}}_{3}({\mathfrak{Q}}_{2})\subseteq{\mathfrak{Q}}_{2}$.
3. The ideal ${\mathfrak{Q}}_{2}$ is generated by the commutation relations

$\begin{array}[]{l}\phantom{.}[u_{i},u_{j}]=0\ \mbox{\rm for }\ i+j<3\
\mbox{\rm or }\ i+j=4;\\\
\phantom{.}[u,u_{3}]=[u_{2},u_{1}]=32i\hbar,\quad[u_{2},u_{3}]=320i\hbar
u_{0},\end{array}$

where $\hbar$ is an arbitrary parameter.

######  Proposition 42.

Quantum $N=2$ KdV hierarchy has the quantum Hamiltonians

${\mathfrak{H}}_{3,5}=-\frac{2}{3}\widehat{{\mathcal{H}}}_{3,5},\qquad{\mathfrak{H}}_{5,5}=-\frac{2}{5}\widehat{{\mathcal{H}}}_{5,5}$

such that $[{\mathfrak{H}}_{3,5},{\mathfrak{H}}_{5,5}]=0$. Here
$\widehat{{\mathcal{H}}}_{3,5},\,\widehat{{\mathcal{H}}}_{5,5}$ are given by
(80).

######  Theorem 43.

For $N=2$ the quantum $N$-th Novikov equation, corresponding to the
derivations $\partial_{t_{1}}$, can be written in the Heisenberg form

$\partial_{t_{1}}(u_{k})=\frac{i}{\hbar}[{\mathfrak{H}}_{3,5},u_{k}]=\begin{cases}u_{k+1},\quad
0\leqslant k\leqslant 2,\\\
32\alpha_{6}-16\alpha_{4}u+5u_{1}^{2}+10u_{2}u-10u^{3},\;k=3.\end{cases}$

The quantum dynamical system ${\mathfrak{C}}_{2}$, corresponding to the
derivations $\partial_{t_{3}}$, can be written in the Heisenberg form

$4\partial_{t_{3}}(u_{k})=\frac{i}{\hbar}[{\mathfrak{H}}_{5,5},u_{k}]=\partial_{t_{1}}^{k}(u_{3}-6u_{1}u),\qquad
k=0,1,2,3.$

###  3.4. Quantum $N=3$ and $N=4$ Novikov hierarchy.

For $N=3$ non-commutative $N$-th Novikov equation has a form

$u_{6}=\mathcal{G}_{8}(u_{0},u_{1},u_{2},u_{3},u_{4}),$ (92)

where

$\mathcal{G}_{8}=7(u_{4}u+uu_{4})+14(u_{3}u_{1}+u_{1}u_{3})+21u_{2}^{2}-21(u_{2}u^{2}+u^{2}u_{2})-28(u_{1}^{2}u+uu_{1}^{2})-\\\
-28uu_{2}u-14u_{1}uu_{1}+35u^{4}-16\alpha_{4}(u_{2}-3u^{2})-64\alpha_{6}u+128\alpha_{8}=\mathcal{G}_{8}^{\dagger}.$
(93)

In ${\mathfrak{B}}_{3}=\mathcal{A}\langle u_{0},u_{1},\ldots,u_{5}\rangle$ the
non-commutative $N$-th Novikov equation is a generator of the two-sided ideal
${\mathfrak{I}}_{3}$.

######  Proposition 44.

The following statements are equivalent
1. The ideal ${\mathfrak{Q}}_{3}$ is ${\mathcal{D}}$-invariant: ${\mathcal{D}}({\mathfrak{Q}}_{3})\subseteq{\mathfrak{Q}}_{3}$.
2. The ideal ${\mathfrak{Q}}_{3}$ is ${\mathcal{D}}_{3}$-invariant: ${\mathcal{D}}_{3}({\mathfrak{Q}}_{3})\subseteq{\mathfrak{Q}}_{3}$.
3. The ideal ${\mathfrak{Q}}_{3}$ is ${\mathcal{D}}_{5}$-invariant: ${\mathcal{D}}_{5}({\mathfrak{Q}}_{3})\subseteq{\mathfrak{Q}}_{3}$.
4. The ideal ${\mathfrak{Q}}_{3}$ is generated by the commutation relations

$\begin{array}[]{l}\phantom{.}[u_{i},u_{j}]=0\ {\rm if}\ i+j<5\ {\rm or}\
i+j=6,\\\
\phantom{.}[u,u_{5}]=[u_{4},u_{1}]=[u_{2},u_{3}]=\eta,\quad[u_{2},u_{5}]=[u_{4},u_{3}]=7\cdot
2\eta u,\\\ \phantom{.}[u_{5},u_{3}]=14\eta
u_{1},\quad[u_{4},u_{5}]=2\eta\hbar(63u^{2}+14u_{2}-8\alpha_{4})\end{array}$

where $\eta\in{\mathbb{C}}$ is an arbitrary parameter.

Setting $\eta=2^{7}i\hbar$, where $\hbar$ is an arbitrary real parameter, we
get.

######  Proposition 45.

Quantum $N=3$ KdV hierarchy has three self-adjoint commuting quantum
Hamiltonians

${\mathfrak{H}}_{3,7}=-\frac{2}{3}\widehat{{\mathcal{H}}}_{3,7}\,,\quad{\mathfrak{H}}_{5,7}=-\frac{2}{5}\widehat{{\mathcal{H}}}_{5,7},\quad{\mathfrak{H}}_{7,7}=-\frac{2}{7}\widehat{{\mathcal{H}}}_{7,7}$

######  Theorem 46.

For $N=3$ the quantum $N$-th Novikov equation, corresponding to the derivation
$\partial_{t_{1}}$, can be written in the Heisenberg form

$\partial_{t_{1}}(u_{k})=\frac{i}{\hbar}[{\mathfrak{H}}_{3,7},u_{k}]=\begin{cases}u_{k+1},&0\leqslant
k\leqslant 4,\\\ \mathcal{{\mathfrak{G}}}_{8},&k=5.\end{cases}$

where

${\mathfrak{G}}_{8}=28u_{3}u_{1}+21u_{2}^{2}+35u^{4}-14(u_{4}\\!-\\!5u_{2}u\\!+\\!5u_{1}^{2})u-16\alpha_{4}(u_{2}\\!-\\!3u^{2})-64\alpha_{6}u+128\alpha_{8}.$

The quantum dynamical systems in ${\mathfrak{C}}_{3}$, corresponding to the
derivations $\partial_{t_{3}}$ and $\partial_{t_{5}}$ can be written in the
Heisenberg form

$\displaystyle 4\partial_{t_{3}}(u_{k})$
$\displaystyle=\frac{i}{\hbar}[{\mathfrak{H}}_{5,7},u_{k}]=\partial_{t_{1}}^{k}(u_{3}-6u_{1}u),$
$\displaystyle 16\partial_{t_{5}}(u_{k})$
$\displaystyle=\frac{i}{\hbar}[{\mathfrak{H}}_{7,7},u_{k}]=\partial_{t_{1}}^{k}(u_{5}-20u_{2}u_{1}-10u_{3}u+30u_{1}u^{2}).$

In the case ${\bf N=4}$ the invariant ideal of quantisation
${\mathfrak{Q}}_{4}$ is generated by the commutation relations
($\eta=2^{9}i\hbar$):

$\begin{array}[]{l}[u,u_{7}]=[u_{6},u_{1}]=[u_{2},u_{5}]=[u_{4},u_{3}]=\eta,\quad[u_{2},u_{7}]=[u_{6},u_{3}]=[u_{4},u_{5}]=18\eta
u,\\\\[5.0pt] [u_{7},u_{3}]=2[u_{4},u_{6}]=36\eta
u_{1},\quad[u_{4},u_{7}]-18\eta
u_{2}=[u_{6},u_{5}]=\eta(198u^{2}+60u_{2}-16\alpha_{4}),\\\\[5.0pt]
[u_{7},u_{6}]=\eta(858u_{1}^{2}-1980u_{2}u-1716u^{3}-54u_{4}+64\alpha_{6}+416\alpha_{4}u),\\\\[5.0pt]
[u_{7},u_{5}]=\eta(396u_{1}u+60u_{3}),\quad[u_{i},u_{j}]=0\;{\rm
if}\;i+j<7\;{\rm or}\;i+j=8.\end{array}$

The corresponding quantum hierarchy takes the form

$2^{2n-2}\partial_{t_{2n-1}}(u_{k})=\frac{i}{\hbar}[{\mathfrak{H}}_{2n+1,9},u_{k}],\qquad
n=1,2,3,4,\ \ k=0,\ldots,7\,,$

where

${\mathfrak{H}}_{2n+1,9}=-\frac{2}{2n+1}\widehat{{\mathcal{H}}}_{2n+1,9},\quad
n=1,2,3,4,$

are commuting self-adjoint Hamiltonians. The first equation of this fine
hierarchy is the $N=4$ Novikov equation written in the Heiseberg form

$\partial_{t_{1}}(u_{k})=\frac{i}{\hbar}[{\mathfrak{H}}_{3,9},u_{k}]=\begin{cases}u_{k+1},&0\leqslant
k\leqslant 6,\\\ \mathcal{{\mathfrak{G}}}_{10},&k=7.\end{cases}$

where

$\displaystyle{\mathfrak{G}}_{10}=512\alpha_{10}-160\alpha_{4}u^{3}+192\alpha_{6}u^{2}+160\alpha_{4}u_{2}u-256\alpha_{8}u+80\alpha_{4}u_{1}^{2}-16\alpha_{4}u_{4}-64\alpha_{6}u_{2}-126u^{5}$
$\displaystyle+420u_{2}u^{3}+630u_{1}^{2}u^{2}-126u_{4}u^{2}-378u_{2}^{2}u-504u_{1}u_{3}u+18u_{6}u+69u_{3}^{2}-462u_{1}^{2}u_{2}+114u_{2}u_{4}+54u_{1}u_{5},$

and the other three equations are ($k=0,\ldots,7$):

$\displaystyle 4\partial_{t_{3}}(u_{k})$
$\displaystyle=\frac{i}{\hbar}[{\mathfrak{H}}_{5,9},u_{k}]=\partial_{t_{1}}^{k}(u_{3}-6u_{1}u),$
$\displaystyle 16\partial_{t_{5}}(u_{k})$
$\displaystyle=\frac{i}{\hbar}[{\mathfrak{H}}_{7,9},u_{k}]=\partial_{t_{1}}^{k}(u_{5}-20u_{2}u_{1}-10u_{3}u+30u_{1}u^{2}),$
$\displaystyle 64\partial_{t_{7}}(u_{k})$
$\displaystyle=\frac{i}{\hbar}[{\mathfrak{H}}_{9,9},u_{k}]=\partial_{t_{1}}^{k}(u_{7}-140u_{1}u^{3}+70u_{3}u^{2}+280u_{2}u_{1}u-14u_{5}u+70u_{1}^{3}-70u_{3}u_{2}-42u_{4}u_{1}).$

There are two interesing observations. In all cases considered in this
section, namely $N=1,2,3,4$:

  * •

The form of the normally ordered quantum N-Novikov equations coincide with the
corresponding classical equations in the commutative case.

  * •

For the first $N$ equations of the quantum $N$-th Novikov hierarchy the
polynomials $\widehat{{\mathcal{H}}}_{2n+1,2N+1},\ n=1,2,\ldots,N$ (80) are
also quantum commuting Hamiltonians. We have shown that these polynomials are
integrals for the non-commutative hierarchy on the free associative algebra
${\mathfrak{B}}_{N}$, i.e.
$\partial_{t_{2k-1}}(\widehat{{\mathcal{H}}}_{2n+1,2N+1})\in\rm{Span}[{\mathfrak{B}}_{N},{\mathfrak{B}}_{N}]$,
where $\partial_{t_{2k-1}},\ k=1,\ldots,N$ . Apparently these derivations map
these polynomials (naturally embedded in ${\mathfrak{C}}_{N}$) into the
corresponding quantisation ideal ${\mathfrak{Q}}_{N}$. In other words

$\partial_{t_{2k-1}}:\widehat{{\mathcal{H}}}_{2n+1,2N+1}\mapsto\rm{Span}[{\mathfrak{B}}_{N},{\mathfrak{B}}_{N}]\bigcap{\mathfrak{Q}}_{N},\qquad
1\leqslant n,k\leqslant N.$

## Acknowledgments.

We would like to thank S.P. Novikov, A.P. Veselov, and V.N. Rubtsov for useful
discussions. A.V. Mikhailov is grateful to the EPSRC (Small Grant Scheme
EP/V050451/1).

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