KdV hierarchies and quantum Novikov's equations

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.


Introduction
We dedicate this paper to the late Decio Levi, in tribute to his profound contributions to the advancement of the theory of integrable systems.His research journey began with the exploration of quantum systems.We believe that our paper, situated at the intersection of classical and quantum integrability and founded on a novel approach to the quantisation problem, will resonate with Decio's interests and legacy.
The problem of quantisation of dynamical systems has a history spanning 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 [16].
]ocnmp[ V M Buchstaber and A V Mikhailov Almost immediately 1 Dirac reformulated Heisenberg's ideas in mathematical form, introduced quantum algebra, and noticed that in the limit → 0 the Heisenberg commutators of quantum observables tend to the Poisson brackets between the corresponding observables in the classical mechanics âb − bâ → i {a, b} [9].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 [9] Dirac stated that "The correspondence between the quantum and classical theories lies not so much in the limiting agreement when → 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 .
Recently, AVM presented a new approach to the problem of quantisation [19].It is suggested to commence 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, can be regarded as a quotient of a free algebra with an equivalent 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 [19] 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 [19] 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 [7].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 [19] to the problem of quantisation of stationary flows of the KdV hierarchy, known as the Novikov equations [5], [8], [15], [22].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 [11], [22].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 2 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 17 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 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 B be an associative C-algebra with the unit 1 and M be a complex linear space.Let ε : B → M be a linear map such that ε(1) = 0.In this paper we introduce the Frobenius-Hochschild algebra F H(B, M).The name and notation are motivated by the fact that the structure of a F H(B, M)-algebra U is given by a skew-symmetric quadratic form Φ on the B-bimodule U with values in M, and this form Φ is a 1-cocycle in the cochain Hochschild complex of the algebra U .Partial cases of the Frobenius-Hochschild algebras are anti-Frobenius algebras.The latter was introduced and developed in connection with the associative Yang-Baxter equation, see [25].In Section 1.2 we describe properties of the F H(B, M)-algebra in the case B = M = A 0 .In terms of the form Φ = σ(•, •) of this algebra, the first N -Novikov integrals are explicitly described in the case of a commutative ring of polynomials A 0 .
The KdV equations with non-commutative matrix variables were introduced in [26], [6].The KdV hierarchies on free associative algebras were studied in [10], [12], [23], [24], [25].In Section 3 we give a description of the integrable KdV hierarchy on a differential graded free associative algebra B 0 = C u, u 1 , . . ., 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 [24].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 B 0 Span([B 0 , B 0 ]) ⊕ D(B 0 ) , see [10], [23], [24].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 α 2n ∈ A, as a generator of the two-sided ideal I N ⊂ B = A u, u 1 . . . .The N -Novikov hierarchy is defined as the canonical projection of the KdV hierarchy to the quotient ring B N = B I N which is free over A and finitely generated.The first system of the hierarchy

written in the form
]ocnmp[ V M Buchstaber and A V Mikhailov of a first order system where D is the derivation of 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 + 8α 4 and in B 1 is represented by the first order system 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 [20].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 F H(B, M)-algebra, where B = B 0 , and M = B 0 /Span([B 0 , B 0 ]).First integrals of the non-commutative N -th Novikov hierarchy are explicitly represented in terms of the form Φ = σ(•, •) of this algebra.Using Lemma 19, we constructed infinitely many algebraically independent first integrals for the non-commutative N -th Novikov equation and its hierarchy.
In Section 4 we consider the quantisation problem for N -th Novikov equation following the method proposed in [19].Let Q N be a commutative graded ring of parameters where |q ω i,j | = i + j + 4 − |ω|, and B N (q) denotes the graded free associative ring B N (q) = Q N u 0 , . . ., u 2N −1 .Having N -th Novikov equation on B N (q), we introduce a differential homogeneous two-sided ideal Q N ⊂ B N (q) generated by the polynomials q ω i,j u ω , 0 i < j 2N − 1, q i,j = 0. ( where , and Q N is invariant with respect to the derivation D. It follows from D(Q N ) ⊂ Q N that the coefficients q i,j , q ω i,j satisfy a system of algebraic equations.In particular, these equation imply that q i,j = 1 for all 0 i < j 2N − 1 (Lemma 33).In the cases N = 1, 2, 3 and 4 we have found out that the all structure constants q ω i,j of the quantisation ideals Q N can be parameterised by one parameter which we denote .In the case N = 1 the computations are presented in full detail in Section 4.2.In this case we have shown that the quantisation ideal for equation (1) is generated by the commutation relation [u 1 , u] = i , which coincides with Heisenberg's commutation relation in quantum mechanics [16], [9].In the case N = 2 we have shown (Proposition 40) that the quantisation ideal Q 2 is generated by six commutation relations The quantum N = 2 KdV hierarchy can be written in the Heisenberg form (Theorem 42) Here the Hamiltonian H 5,3 ∈ C 2 for the Novikov equation coincides with the first integral of weight 8 in the commutative case, assuming that all monomials are normally ordered, while the Hamiltonian H 5,5 ∈ C 2 requires a quantum correction (Proposition 41).These Hamiltonians commute with each other [H 5,3 , H 5,5 ] = 0. We conclude Section 4 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 [19] does not assume any Hamiltonian structure of the noncomutative dynamical system, nevertheless we present the quantum equations in the Heisenberg form 2 Novikov's equations and the corresponding finite KdV hierarchies.
2.1 Lie algebra of evolutionary differentiations.
Consider a graded commutative differential polynomial algebra where D is a derivation of C[u 0 , u 1 , . ..] such that D(u k ) = u k+1 , k = 0, 1, . . .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 is a derivation in A 0 .Its action X : A 0 → A 0 is well defined, since any element a ∈ 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 C linearity and the Leibniz rule are obviously satisfied.For example, partial derivatives ∂ ∂u i , i = 0, 1, . .., are commuting derivations in 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 ]ocnmp[ V M Buchstaber and A V Mikhailov (4) can be expressed as f n = D n (f ) in terms of one element f ∈ A 0 , which is called the characteristic of the evolutionary derivation.We will use notation 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 Der A 0 .Indeed, where [f, g] ∈ A 0 denotes the Lie bracket which is bi-linear, skew-symmetric and satisfying the Jacobi identity.Thus A 0 is a Lie algebra with Lie bracket defined by (6).Let a(u, . . ., u n ) be a non-constant element of A 0 .Then X f (a) can be represented by a finite sum where is the Fréchet derivative of a(u, . . ., 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 An evolutionary derivation X f we identify with the partial differential equation Following [25] we define symmetries of (10).
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 ∂ τ which commutes with ∂ t .

Frobenius-Hochschild algebras.
We shall assume that u is a smooth function u = u(t 1 , t 3 , . . ., t 2k−1 , . ..) of graded variables t 2k−1 , k = 1, 2, . .., where |t 2k−1 | = 1−2k.The variable t 1 we will identity with x.We use abbreviated notations for partial derivatives ∂u Let us define a differential operator of order m as a finite sum of the form Differential operators act naturally on the algebra A 0 .
The set of differental graded operators and the set of graded formal differential series are non-commutative associative algebras.In this algebra, multiplication is defined by the composition of series using the formula reflecting the Leibniz rule.Here For any two elements A, B ∈ A D 0 we have the commutator [A, B] = AB − BA.For instance, for any a ∈ A 0 , the formulas are fulfilled: Proof.
2) For any elements A, B, C of any associative algebra, the identity holds.Therefore, for a ∈ A 0 and B, C Applying the operator "res" to (17) and using already proved statement 1), we obtain the proof of statement 2).
Let B be some associative C-algebra with the unit 1 and M some complex linear space.Let a linear mapping ε : B → M be given such that ε(1) = 0. Definition 4.An associative C-algebra U with unit 1 will be called a Frobenius-Hochschild algebra over (B, M) (briefly F H(B, M)-algebra) if: i) The algebra B is a subalgebra of U , and hence U is a two-sided B-module.
ii) The bilinear mapping Φ(•, •) : U ⊗ C U → M is defined such that: Proof.Let us substitute C = 1 in (18).Then, according to item 1) of the Definition 4, we obtain a proof of assertion a).If we substitute C = a in (18) then, according to item 1) of the Definition 4, obtain a proof of assertion b).
Theorem 6.The algebra A D 0 is a F H(A 0 , A 0 )-algebra in which the bilinear form Φ = σ : Then aA ∈ A D 0 for any a ∈ A 0 and therefore the algebra A D 0 is a left A 0 -module with respect to the embedding ε : A 0 → A D 0 : a → aD 0 .According to (13), the structure of the right A 0 -module is given by the formula According to (19), we obtain σ(ε(a), A) = σ(aD 0 , A) = 0. Thus item 1) of condition ii) of the Definition 4 has been verified.
The proof of item 2) of condition ii) is based on two lemmas, which are of independent interest: Proof.Forms D σ(•, •) and res [•, •] are bilinear so it suffices to proof the relation: According to the condition of Theorem 6, for any a, b ∈ A 0 we have But according to item 2) of Lemma 3 Therefore, it suffices to proof the relation (21) in the case a = 1.But in this case we have: Lemma 7 is proved.
]ocnmp[ V M Buchstaber and A V Mikhailov The monomials form an additive basis of the graded algebra A 0 = C[u 0 , u 1 , . ..].We will consider A 0 as a graded algebra A 0 = C ⊕ A 0 , where A 0 = ⊕ m A m 0 and A m 0 is a graded finite-dimensional C-linear subspace in A 0 with an additive basis {u ξ , |u ξ | = m}.
The vectors of the space A m 0 are called homogeneous polynomials of weight m.Let us introduce an ordering of the multiplicative generators of the algebra A 0 : Then a strict order is defined in the monomial basis {u ξ } of the space A m 0 for each m > 0. This order is induced by the lexicographic order of the sequences ξ.
Lemma 8.For any m > 0 the homomorphism D : The composition of linear homomorphisms maps the ordered set of monomials u ξ ∈ A m 0 into the ordered set of monomials u ξ ′ ∈ A m+1 0 .We will show that this mapping is monotone and thus we obtain that the homomorphism D is a monomorphism for m > 0. Let We now continue the proof of Theorem 6.It remains to prove that item 2) of condition ii) of Definition 4 is satisfied, i.e. that relation (18) is true.
Let A, B, C ∈ A D 0 .Take the residue res of the left side of equality (16).Then, according to Lemma 7, we obtain: Since according to Lemma 8 the operator D is the monomorphism on non-constant series, we obtain that relation (18) is true.Theorem 6 is proved.
For A = i m a i D i , a m = 0, we put A = A + + A − where A + = 0 if m < 0, and Corollary 10.
Lemma 12.A homogeneous formal series where I 1,n ∈ A 0 are homogeneous polynomials of the weight n + 1, satisfies the equation Proof.Consider the equation We obtain Using (13), we get (26).

Let us define a sequence of differential operators
where
Let J = u, u 1 , . . .⊂ A 0 be the two-sided maximal ideal generated by u, u 1 , . . .Proposition 14.For k ∈ N the following formula holds: where Proof.By definition, ρ 2n = res L where [x, a] = 0. We have Therefore, the desired coefficient is Formula (26) implies that Using formulas (30) and (32), we obtain by induction that ]ocnmp[ V M Buchstaber and A V Mikhailov Examples: It is easy to show that [L 2k−1 , L] = 0 and therefore the commutator is the operator of multiplication on the function 2D(ρ 2k ).
Definition 15.The KdV hierarchy is defined as an infinite sequence of differential equations Examples: and so on.The partial derivatives ∂ t 2k−1 can be extended to derivations of the algebra Therefore the KdV hierarchy can be written in the form of Lax's equations It can be shown, that the derivations ∂ t 2k−1 commute with each other [25], and thus the KdV hierarchy is a system of compatible equations.It follows from and therefore, Let's put According to Corollary 10, we obtain: Taking the residue from the equation 36, we get Thus {ρ 2n , n ∈ N} is a sequence of common conserved densities for the infinite KdV hierarchy (34), and σ 2k−1,2n−1 are homogeneous differential polynomials, On the algebra A 0 the evolutionary derivations ∂ t 2k−1 are represented by commuting derivations In particular The N-th Novikov hierarchy.
Let us define a symmetry ∂ τ of the KdV equation taking a linear combination with constant coefficients of the first N members of the KdV hierarchy (34) Let us define a polynomial In (42) we assume that ρ 0 = 1 and α 2N +2 is a constant parameter of weight |α 2N +2 | = 2N + 2. The polynomial F 2N +2 (see ( 42)) is homogeneous of weight 2N + 2. Let us restrict ourselves with solutions of the KdV hierarchy which are invariant with respect to the symmetry (41).It implies that It follows from (33) that equation (43) can be resolved with respect to the variable u 2N and written in the form ]ocnmp[ V M Buchstaber and A V Mikhailov 44) is called N -th Novikov equation.Since ρ 2n ∈ A 0 , these equations depend linearly on α 4 , . . ., α 2N +2 .For example: , the N -th Novikov equation is an ordinary differential equation of the 2N -th order for the function u = u(x).
Let I N = (F 2N +2 ) ⊂ A be a differential ideal generated by the polynomial F 2N +2 and the D derivatives.For any element of A the canonical projection is the result of the elimination of variables u k , k 2N , using equation ( 44) and equation Proposition 16.The ideal I N is invariant with respect to evolutionary derivations Proof.Indeed, it follows from ( 7), (38), and (40) that In C 2N there are N compatible systems of N ordinary differential equations which we will call N -th Novikov hierarchy.In this case, the parameters α 2k are assumed to be fixed complex numbers.In the hierarchy (46), system with k = 1, s = 0, . . ., N − 1 represents the N -th Novikov equation (44) as a first order system of 2N ordinary differential equations.
Proof.It follows from (38) that where and according Proposition 17 we get one first integral The hierarchy consists of two compatible systems in which the first one is the N = 2 Novikov equation 3 KdV hierarchy and Novikov equations on free associative algebra.
3.1 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 B 0 = (C u 0 , u 1 , . . ., D) with infinite number of noncommuting variables (see for example, [24], [25]).Algebra B 0 has monomial additive basis induced by the grading of the variables u k , |u k | = k + 2 for any k 0 and therefore The construction of the hierarchy is similar to the commutative case, although one has to take care on the order of the variables, since Starting with the operator L = D 2 − u, one can find its square root by the formula where I 1,n ∈ B 0 are non-commutative polynomials.It follows from the proof of Lemma 2 that formula (26) for the recursive calculation of the polynomials I 1,n is also applicable in the case of a free associative algebra B 0 .Now the initial segment of the series L has the form Similarly to the commutative case, we introduce fractional powers L 2k−1 and polynomials ̺ 2k = res L 2k−1 .From the identity L 2k+1 = LL 2k−1 follows a formula for the recursive calculation of the polynomials ̺ 2k+2 .It follows from the proof of the formulas (30) and (32) that they are applicable in the case of a free associative algebra B 0 .However, expressions for ̺ 2k ∈ B 0 are different from expressions for ρ 2k ∈ A 0 , k 3, and so on.
Definition 18.The compatible system of equations on the free associative algebra B 0 is called the KdV hierarchy (similar to the commutative case (34)).
]ocnmp[ V M Buchstaber and A V Mikhailov Equations of the KdV hierarchy define the commuting evolutionary derivations D 2k−1 of B 0 .Their action on the variables u n is given by and it can be extended to B 0 by the linearity and the Leibniz rule.
We have In the non-commutative case res [A, B] is not any more in the image of the derivation D and Lemma 7 should be modified.Let us introduce the algebra Definition 19.Let us introduce the homogeneous skew-symmetric bilinear over C form Proof.The statement of this lemma is verified by directly calculating the value of ∆(aD n , bD m ).

Corollary 21.
Let where In the non-commutative case the definition of densities of local conservation laws has to be modified, since Here Span[B 0 , B 0 ] is a linear subspace generated by all commutators of elements from B 0 .

Frobenius-Hochschild algebras over free associative algebra.
Denote by ξ k = (j 1 , . . ., j k ) sequences of non-negative integers of length k 1.Let We will consider B 0 as a graded algebra B 0 = C ⊕ B 0 , where B 0 = ⊕ m B 0,m , m 2, and B 0,m is a graded finite-dimensional C-linear space with an additive lexicographically by indices ordered monomial basis For example, {u 3 , u 1 u, uu 1 } is a monomial and lexicographically ordered basis 0 .Let ξ k = (j 1 , . . ., j k ) be a multindex of the monomial u ξ k and T k be a generator of the cyclic permutation group of order k: T 1 (ξ 1 ) = ξ 1 and T k (ξ k ) = (j 2 , . . ., j k , j 1 ), k 2. Let us denote by T (ξ k ) the maximal index set in T (ξ k ) = max ≻ {ξ k , T k (ξ k ), . . ., T k−1 k (ξ k )} with respect to the lexicographic ordering.We define the linear homomorphism T : B 0,m → B 0,m by its action on the monomial basis elements T (u ξ k ) = u T (ξ k ) .
For example, T (u 1 u) = T (uu 1 ) = u 1 u.Proof.It follows directly from the definition that T = T 2 .The properties of T follow from the following facts: It follows from Proposition 22 that the projector T gives the splitting of the exact sequence If there are at least two distinct elements in the set ξ k , then D(u ξ k ) is a sum of monomials with the leading monomial u j 1 +1, * u j 2 , * • • • u j k, * .Thus, in a strictly ordered basis, the homomorphism D : B 0,m → B ♮ 0,m+1 is given by an upper triangular matrix with a non-zero diagonal.It induces the monomorphism D : B ♮ 0,m → B ♮ 0,m+1 .Following the proof of Theorem 6 and using Lemma 20, it is easy to complete the proof of Theorem 23.
which is called the non-commutative N -th Novikov equation. ]ocnmp[

V M Buchstaber and A V Mikhailov
There are infinitely many algebraically independent first integrals (in the sense of Definition 25) given by (69).For example, it follows from (69) that in B ♮ 1 We have and 3.4 Self-adjointness of the KdV and N-Novikov hierarchies.
Let B A = A u, u 1 , . . . .The set of differential operators and the set of differential formal series are non-commutative associative algebras in which multiplication is defined by formula (13).According to formula (13), a conjugation anti-automorphism † : is defined on the ring B D A .
Lemma 26. 1.The operator † is uniquely defined by the conditions

On the ring B
with parameters, which are in the centre of the ring. Let q) be a two-sided D = D 1 differential homogeneous ideal generated by the polynomials Let us consider the graded associative algebra C N = B N (q)/Q N and the ring epimorphism B N (q) → C N preserving the grading.
Definition 31.The ideal Q N is called the Poincaré-Birkhoff-Witt ideal (briefly, the PBW-ideal) if the image of the set of monomials u ω , ω ∈ Z 2N , forms a non-degenerate additive basis in the Q-module C N .
Definition 32.The ideal Q N is a quantisation ideal of the N -th Novikov equation if: 1. it is the PBW-ideal; 2. it is invariant with respect to the derivation D.
Condition 2 of Definition 32 reduces to a system of polynomial algebraic equations in Q N .
Lemma 33.Let Q N ⊂ 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 D to the polynomial p 2N −2,2N −1 we get where Let us assume that q i,j = 1 for all i < j, such that i + j k.For a polynomial p i,j with i < j and i + j < k we get By the induction assumption q i,j+1 = 1.Therefore, 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 Lm(f i,j ) ≺ u j+1 u i in the additive basis of C N .It follows from D(p i,j ) ∈ Q N that q i,j = 1.
1.The relation ]ocnmp[ V M Buchstaber and A V Mikhailov holds in the ring C N , where q ω i,j u ω , q ω i,j ∈ Q N .

For any
then the derivation D induces a well defined derivation ∂ t 1 on the quotient algebra B N (q)/Q N and a quantum dynamical system defined by the quantum N -th Novikov equation where Theorem 35.The ideal Q N is the quantisation ideal of the N -th Novikov equation if and only if the set of polynomials h ij ∈ C N is a solution of the following systems in the ring C N :

I. the system of algebraic equations linear in h
for all triples (i, j, k), 0 i < j < k 2N − 1;

II. the system of differential equations linear in h
where Proof.In [18], V. Levandovskyy obtained necessary and sufficient conditions on polynomials p i,j of the form (83) under which the ideal Q N is the BPW-ideal.Under the additional condition q i,j = 1 (see Lemma 33), Lemma 2.1 from [18]  Therefore, the polynomial ξ must satisfy the equation Accordingly to formula (83), the solution to this equation should be sought in the form: Substituting expression (93) into equation (92) and using the PBW-basis in C 2 , we obtain that ξ = 0 in C 2 .Then the polynomial ξ ∈ B 2 (q) must belong to the ideal Q 2 , but this is possible only when [u, u 1 ] = 0.Under condition [u, u 1 ] = 0, the system of differential equations (85) in the case N = 2 takes the form h 01 = 0; h 02 = 0; h 12 + h 03 = 0;

The ideal
3. The ideal Q 2 is generated by the commutation relations [u i , u j ] = 0 for i + j < 3 or i + j = 4; [u, where is an arbitrary parameter.The quantum dynamical systems in C 3 , corresponding to the derivations ∂ t 3 and ∂ t 5 can be written in the Heisenberg form In the case N = 4 the invariant ideal of quantisation Q 4 is generated by the commutation relations (η = 2 9 i ): operator L = D 2 − u, D = d dx .The KdV hierarchy defines evolutionary derivations in the graded algebra A 0 = C[u 0 , u 1 , u 2 , . ..] 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 , . . .as smooth functions u k = u k (t 1 , t 3 , . ..) of graded variables t 2k−1 , k ∈ 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 I N in the graded ring A = A[u, u 1 , . ..],where A is a commutative algebra of graded parameters α 4 , α 6 , . . ., α 2N +2 where the weights |α 2n | = 2n.The Proposition 16 shows that the KdV hierarchy induces the finite N hierarchy of integrable ordinary differential polynomial equations on the quotient ring A I N which is called N -th Novikov hierarchy.

Definition 2 . 3 . 1 ) 2 )
For a formal series A ∈ A D 0 the coefficient a −1 of the term a −1 D −1 is called the residue of this series A and denoted by res A. Lemma For any B ∈ A D 0 and a ∈ A 0 we have res [a, B] = 0.For any a ∈ A 0 and B, C ∈ A D 0 res [aB, C] = res [B, Ca].

Theorem 11 .
The form σ(•, •) is given in terms of the operation res by the recursive formula σ(A, BD) = σ(DA, B) − res AB (25) with the initial condition σ(A, bD) = −res Ab for any A ∈ A D 0 and b ∈ A 0 .

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)

Corollary 24 . 3 . 3
The algebra B D 0 is the F H(B 0 , B ♮ 0 )-algebra with the bilinear form σ(A, B) (62).Proof.It follows immediately from Theorem 23 and the fact that σ(A, B) − σ(A, B) ∈ Span[B 0 , B 0 ] for any A, B ∈ B D 0 .Non-commutative N-th Novikov equation and its hierarchy.Let B = (A u, u 1 , . . ., D) be the differential ring, where A = C[α 4 , . ..],D(u k ) = u k+1 , D(α 2n ) = 0, and α 2n are commuting parameters, i.e centre elements of B. Similar to the commutative case, we fix a positive integer N and define a homogenous polynomial F 2N +2 ∈ B: Let I N ⊂ B be a two-sided differential ideal generated by the polynomials F 2N +2 and D k (F 2N +2 ), k ∈ N. The quotient ring B N = B/I N is a graded finitely generated free ring B N = A u 0 , u 1 , . . ., u 2N −1 over A. Similar to Proposition 16 we can show that equations of the hierarchy (59) are all compatible, therefore D 2k−1 (I N ) ⊂ I N .The derivations D 2k−1 induce derivations D 2k−1 of the quotient algebra B N .The equation F 2N +2 = 0 can be written in the form A the operators D and † commute, i.e. (D(a)) † = D(a † ) for any a ∈ B A .