Quantisations of the Volterra hierarchy

In this paper we explore a recently emerged approach to the problem of quantisation based on the notion of quantisation ideals. We explicitly prove that the nonabelian Volterra together with the whole hierarchy of its symmetries admit a deformation quantisation. We show that all odd-degree symmetries of the Volterra hierarchy admit also a non-deformation quantisation. We discuss the quantisation problem for periodic Volterra hierarchy including their quantum Hamiltonians, central elements of the quantised algebras, and demonstrate super-integrability of the quantum systems obtained. We show that the Volterra system with period $3$ admits a bi-quantum structure, which can be regarded as a quantum deformation of its classical bi-Hamiltonian structure.


Introduction
The problem of quantisation has a century long history. In 1925, inspired by Heisenberg's commutation relations between coordinates and momenta [1], namely, q npm −p mqn = i δ n,m ,q nqm −q mqn = 0,p npm −p mpn = 0, n, m = 1, . . . , N, Dirac proposed the concept of quantum algebra and noticed that in the limit → 0 the commutators of observables are proportional to their Poisson brackets in classical mechanics [q n ,p m ] → i {q n , p m }.
He raised the issue of consistency of the commutation relations (1) with each other and with the equations of motion for a finite Plank constant = 0 [2]. In fact, Dirac proposed the problem of non-commutative deformations of multiplication on Poisson manifolds that is presently an active research area. Important results in this direction have been obtained by Kontsevich [3]. Witten, in his recent lectures [4], pointed out that due to "the operator ordering problem, there is no natural, general procedure to quantize a classical system", and described some partial remedies to this problem. The general problem of quantisation is still open.
Recently, a fresh approach to the quantisation problem was proposed in [5]. It is proposed to start from a dynamical system defined on a free associative algebra A with a finite or infinite number of multiplicative generators. The dynamical system defines a derivation ∂ t : A → A. By quantisation it is understood a reduction of the dynamical system on A to the system defined on a quotient algebra A I = A I over a two-sided ideal I ⊂ A satisfying the following properties: (i) the ideal I is ∂ t -stable, that is, ∂ t (I) ⊂ I; (ii) the quotient algebra A I admits an additive basis of normally ordered monomials.
In [5] an ideal satisfying the above two conditions is called a quantisation ideal , and A I is called a quantised algebra.
The condition (i) is crucial. The reduction of a dynamical system corresponding to the derivation ∂ t to the quotient algebra A I is well defined if and only if the ideal is ∂ t -stable.
The second condition (ii) enables one to define commutation relations between any two elements of the quotient algebra and uniquely represent elements of A I in the basis of normally ordered monomials (similar to a normal ordering in quantum physics). Finitely generated algebras, admitting a Poincaré-Birkhoff-Witt basis, and their quotients, satisfy the condition (ii). They have a wide range of applications, and share some properties with the commutative polynomial rings (see [6,7] and references in).
Any finitely generated associative algebra can be presented as (is isomorphic to) a quotient of a free associative algebra over a suitable two-sided ideal. For example, Dirac's quantum algebra is a quotient of the free algebra C q 1 , p 1 , . . . , q N , p N over the two-sided ideal generated by the commutation relations (1).
We emphasise that quantisation proposed in [5] guarantee the consistency of the "commutation relations" with each other and with the equations of motion (resolving the issue raised by Dirac) and the associativity of the non-commutative multiplication in the quantised algebra (which potentially could be an issue in the deformation quantisation). This new approach also results in examples of non-deformation quantisations.
In order to apply this method of quantisation to a classical dynamical system with commutative variables one needs to lift it to a system on a nonabelian free associative algebra. Such lifting is not unique (on the quantum level it has been noted already by Dirac [2], and highlighted by Witten in his lectures [4]). The guiding principle here is to preserve the most important properties of the classical system in the lifted one. For example, integrable systems admit hierarchies of symmetries and we would like to have this property for the corresponding systems defined on a free associative algebras and for the quantised systems as well. Fortunately many integrable systems admit such liftings [8,9,10,11,12], and can be quantised by the method proposed in [5]. Recently, the hierarchies of stationary Korteweg de-Vries equation and Novikov's equations have been quantised using the method of quantisation ideals [13].
In this paper we study the quantisation problem for the integrable nonabelian Volterra system ∂ t 1 (u n ) = ̺K (1) , K (1) = u n+1 u n − u n u n−1 , n ∈ Z (2) and its hierarchy of symmetries. Here ̺ ∈ C is a constant which can be set to be equal to 1 by the re-scaling u n → ̺u n . In the classical (commutative) case system (2) was introduced by Zakharov, Musher and Rubenchik for the description of the fine structure of the spectra of Langmuir oscillations in a plasma [14]. Its integrability and Lax representation were discovered by Manakov [15] and independently by Kac and van Moerbeke [16]. The nonabelian version of the system (2), with variables u n (t 1 ) taking values in a free associative algebra, was studied by Bogoyavlensky [17].
In the new approach the quantisation problem for equation (2) reduces to the problem of finding two-sided ideals in the free associative algebra A = C u n ; n ∈ Z generated by an infinite number of non-commuting variables such that the above conditions (i) and (ii) are satisfied. It is obvious that the ideal I generated by the infinite set of polynomials I = u n u m − ω n,m u m u n ; n, m ∈ Z, ω n,m ∈ C * (4) satisfies the condition (ii) for any choice of the parameters ω n,m = ω −1 m,n . In [5] it was stated that the ideal I satisfies the condition (i) if and only if Thus the quantisation ideal suitable for the Volterra system (2) is leading to the commutation relations in the quotient algebra A I a . It was verified by direct computations that the ideal I a is invariant with respect to derivations defined by a few first symmetries of the Volterra hierarchy and conjectured that it is also true for the whole hierarchy. In this paper we give an explicit proof for the above conjecture (Theorem 9). The ideal I a corresponds to a deformation quantisation. In the limit ω → 1 it leads to the classical commutative case.
It was claimed in [5] that the cubic symmetry of the Volterra system, equation (3), admits two distinct quantisations ideals of the form (4). The first one coincides with I a defined by (5), while the second one is Note that the quantisation corresponding to the ideal I b is not a deformation of a commutative or Grassmann algebra. It is a new and non-deformation quantisation of equation (3) with the commutation relations u n u n+1 = (−1) n ωu n+1 u n , u n u m + u m u n = 0 if |n − m| 2, n, m ∈ Z in the quotient algebra A I b . The ideal I b given by (7) is not invariant with respect to the Volterra system (2) and thus it is not suitable for its quantisation. In [5] it was claimed that the ideal I b is invariant with respect to a first few odd degree symmetries of the Volterra equation. In this paper we prove that the ideal I b (7) is a quantisation ideal for all odd degree members of the Volterra hierarchy (Theorem 14).
In the quantum theory we replace real valued commutative variables u n by Hermitian elements. Their commutation relations are defined by the quantisation ideal, which should be stable with respect to the Hermitian conjugation (Definition 3). In the case of the ideals I a and I b , it implies that ω = e 2i , where is an arbitrary real parameter, an analogue of the Plank constant, and i 2 = −1. Moreover, in the quantised equations of the Volterra hierarchy, we should introduce the factors e iℓ which make the right-hand side of the equations self-adjoint, that is, . . , u n−ℓ ), ℓ = 1, 2, . . . , n ∈ Z.
In the algebra A Ia with commutation relations (6) the quantised Volterra equation and its symmetry can be represented in the Heisenberg form where In the algebra A I b with commutation relations (8), the first member of the quantised Volterra subhierarchy of odd degree symmetries has the same Heisenberg form (11). Moreover, in the case of the algebra A I b we have H 2 = H 2 1 , which is not true for the algebra A Ia . The quantisation of the Volterra system was studied by Volkov and Babelon in the frame of the quantum inverse scattering method [18,19]. In the paper by Inoue and Hikami [20], the commutation relations (6), as well as a first few Hamiltonians of the classical and quantum Volterra hierarchy were found using ultra-local Lax representation and R-matrix technique. Our alternative approach does not rely on the existence of a Lax or Hamiltonian structures, and it enables us to reproduce the results presented in [20] and to find a non-deformation quantisation (8) for odd degree members of the Volterra hierarchy which is new and rather surprising.
The Volterra equation and its hierarchy admit periodic reductions with arbitrary positive integer period M ∈ N. The periodic reduction is the identification u n+M = u n for all n ∈ Z. It reduces the infinite system of equations (2) to a system of M equations on a finitely generated free algebra A M = C u 1 , . . . , u M . The problem of quantisation of the periodic Volterra hierarchies is discussed in Section 4. In particular, we show that the Volterra system with period 3 admits bi-quantum structure, which is a quantum analogue of its bi-Hamiltonian structure in the classical case. In the case M = 4 we obtain three possible quantisations, and show that the obtained quantised systems are super-integrable, whose first integrals and central elements are explicitly presented.

Integrable nonabelian Volterra hierarchy
In this section we introduce some basic notations required for this paper, and present the Volterra hierarchy on a free associative algebra in an explicit form.
Let A = C u n ; n ∈ Z be a free associative algebra generated by an infinite number of noncommuting variables. There is a natural automorphism S : A → A, which we call the shift operator, defined as Thus A is a difference algebra. Let T denote the antiautomorphism of A defined by The involution T is a composition of the reflection in the alphabet index u k → u −k and the transposition of the monomials. For example: A derivation D of the algebra A is a C-linear map satisfying Leibniz's rule Thus a derivation D can be uniquely defined by its action on the generators and D(α) = 0, α ∈ C.
A derivation D is called evolutionary if it commutes with the automorphism S. An evolutionary derivation is completely characterised by its action on the generator u (we often write u instead of u 0 ), that is, Thus it is natural to adopt the notation D a , such that D a (u) = a, for an evolutionary derivation with the characteristic a. Assuming that the generators u k depend on t ∈ C we can identify an evolutionary D a with an infinite system of differential-difference equations Therefore we can say that ∂ t (u) = a defines a derivation of A.
The Volterra system (2) defines the derivation ∂ t 1 : A → A, which commutes with the automorphism and anti-commute with the involution T , i.e., The differential-difference system (3) defines another evolutionary derivation ∂ t 2 commuting with S and anti-commuting with T . Evolutionary derivations commuting with ∂ t 1 are symmetries of the Volterra system. It can be straightforwardly verified that [∂ t 1 , ∂ t 2 ] = 0 and thus equation (3) is a symmetry of the Volterra system.
It is well known that the Volterra system has an infinite hierarchy of commuting symmetries. They can be found using Lax representations both in commutative [15] and non-commutative [17] cases, or the recursion operators [21,12]. Remarkably, the explicit expressions for generalised symmetries of the Volterra system (2) can be presented in terms of a family of nonabelian homogeneous difference polynomials [12], which is inspired by the polynomials in the commutative case discovered in [22,23].
Let us assume that the generators u k of the free associative algebra A depend on an infinite set of "times" t 1 , t 2 , . . . . It follows from [12] that the hierarchy of commuting symmetries of the Volterra system (2) can be written in the following explicit form where the (noncommutative) polynomials X (ℓ) are given by explicit formulae Here →ℓ j=1 denotes the order of the values j, from 1 to ℓ in the product of the noncommutative generators u λ j +1−j . For example, we have X (1) = u and Note that T (X (ℓ) ) = X (ℓ) , and thus we have T · ∂ t ℓ = −∂ t ℓ · T for all ℓ. Clearly, we get the Volterra equation (2) when ℓ = 1 and the system (3) when ℓ = 2.

Quantisation ideals of the Volterra equation and its symmetry
In this section, we prove the statements on quantisation ideals for the Volterra equation (2) itself and its symmetry (3) stated in [5].
Let I ⊂ A be a two-sided ideal generated by the infinite set of polynomials f i,j : where ω i,j ∈ C * are arbitrary non-zero complex parameters. Given an ideal I, we denote the projection on the quotient algebra by by π I : A → A/I. The quotient algebra A I has an additive basis of standard normally ordered monomials Indeed, in A I any polynomial can be represented in this basis by recursive replacements u n u m → ω n,m u m u n if m > n in the monomials. Thus the condition (ii) for the ideal I is satisfied. The condition (i) imposes constraints on the structure constants ω n,m of the ideal. Denoting ω 0,1 = ω, we arrive to the commutation relations (6) and the ideal I a given by (5).
Proof. Let us differentiate f i,j (i < j) by the derivation ∂ t 1 associated to the Volterra equation (2). We have We project this equation on the quotient algebra and require where we use the convention ω i,i = 1. When j > i + 2, the four monomials u j+1 u j u i , u j u i u i−1 , u j u i+1 u i and u j u j−1 u i are linearly independent. Thus π I (∂ t 1 (f i,j )) = 0 if and only if all their coefficients vanish since ω i,j = 0. This leads to Hence we must have ω i,j = 1 whenever i + 1 < j. Using this result, it follows from (17) that This implies that all the ω i,i+1 are equal to each other. Let ω = ω i,i+1 . It remains to check that (17) is valid for j = i + 1. Indeed, and we proved the statement.
Thus, equation (3) admits the same quantisation A I a (5) as the Volterra system. Additionally, it admits the quantisation with the ideal I b (7), which is not invariant with respect to the Volterra system (2). The latter quantisation is not a deformation of a commutative system.
Proof. We differentiate f i,j (i < j) by the derivation ∂ t 2 defined by equation (3) and project on the quotient algebra. When i + 2 j we have where we use the convention ω i,i = 1. If i + 3 < j all monomials in (18) are distinct and one deduces from becomes We now look at ∂ t 2 (f i,i+2 ). Substituting j = i + 2 into (18), we get which vanishes if and only if ω i,i+1 = ǫω i+1,i+2 . Combining all the constraints obtained on ω i,j , we obtain the two cases listed in the statement. Finally, we check Thus we complete the proof.
In section 5 we will show that every member of the Volterra hierarchy (12) admits the quantisation A I a (Theorem 9) and that every even member of the Volterra hierarchy also admits the quantisation A I b (Theorem 14).
In the classical commutative case the variables u n are usually assumed to be real valued. Thus, in the quantum case they should be presented by self adjoint operators with respect to the Hermitian conjugation †.
Definition 3. The Hermitian conjugation † in algebra A is defined by the following rules whereᾱ is the complex conjugate of α ∈ C.
The algebra A is Z 2 -graded as a linear space. It can be represented as a direct sum of self-adjoint and anti-self-adjoint subspaces The Hermitian conjugation † can be extended to the quantised algebra A I if the ideal I is †-stable: Proof. Indeed, in the case of the ideal I a we have In the case for I b , the proof is similar.
It suggests to represent ω = q 2 , q = e i , where ∈ R is a real constant (an analog of the Plank constant). Thus (u n+1 u n ) † = u n u n+1 = q 2 u n+1 u n . The quantum Volterra hierarchy, which is consistent with the condition u † n = u n , can be presented in the form Finally, we present the Volterra system and its first symmetry in the Heisenberg form in the quotient algebras. In the algebra A I a with commutation relations (6) the Volterra equation (2) and its symmetry (3) can be represented in the Heisenberg form where H 1 and H 2 are self-adjoint algebraically independent and commuting Hamiltonians [H 1 , The quantisation A I b with commutation relations (8) also enables us to present equation (3) in the Heisenberg form Note that in the quantised algebra A I b we have H 2 = H 2 1 and H † 2 = H 2 .

Periodic Volterra hierarchy
In the Volterra system (2) we can assume that the function u n (t 1 ) is periodical in n with an integer period M ∈ N, that is, u n = u n+M , n ∈ Z. In this case the infinite dimensional system (2) reduces to the M -dimensional dynamical system on The ideal I M is obviously stable with respect to evolutionary derivations. We can take u n , n = 1, . . . M as canonical representatives of the cosets u k +I M , k ∈ Z. The algebra A M is a difference algebra with the induced automorphism S(u k ) = u (k+1) mod M of order M .
The hierarchy of symmetries (12) of the Volterra system (2) reduces to the hierarchy of symmetries of the M -periodic system provided we count the subscript k in u k modulo M . The cases M = 1, 2 lead to trivial equations.
In the case M = 3 the periodic Volterra system takes the form It has an infinitely hierarchy of commuting symmetries: For any M the nonabelian Volterra hierarchy has a common first integral H = M k=1 u k .
In the case of the finitely generated free algebra A M we consider more general inhomogeneous ideals I M ⊂ A M (than (4)) generated by the polynomials f i,j : where ω i,j = 0, ω i,j , σ r i,j , η i,j ∈ C and we use Einstein summation convention, namely σ r i,j u r denotes M r=1 σ r i,j u r . In this section, we explore the quantisation problem for periodic reductions of the Volterra system and its cubic symmetry.

Quantisation of the periodic Volterra system
Similarly to what we did in Section 3, we are able to prove the following statement for the periodic Volterra equation: The constants α, β, γ, η ∈ C, α = 0 are arbitrary.
Proof. When M = 3, the ideal I 3 is generated by three polynomials f 1,2 , f 1,3 and f 2,3 . We differentiate them by the derivation ∂ t 1 associated to the Volterra equation (22) and project it on the quotient algebra. We have In the same way, we compute π I 3 (∂ t 1 (f 2,3 )) and π I 3 (∂ t 1 (f 1,3 )). If I 3 is preserved under the derivation ∂ t 1 , all coefficients in these expressions should vanish, which leads to an algebraic system for ω i,j , σ r i,j , η i,j , 1 ≤ i < j ≤ 3 and r ∈ {1, 2, 3}. The only nontrivial solution of this system is which is the ideal presented in the statement by setting ω 1,2 = α, σ 1 1,2 = β and η 1,2 = η. The proof of the statement for the case when M = 4 is similar and we do not present it here. Let us now prove the last part of the statement concerning the case M 5. The condition M 5 implies that u n+2 , u n+1 , u n , u n−1 , u n−2 are algebraically independent in A M /I M for all n ∈ Z. In the quotient algebra A M /I M , π I M (∂ t 1 (f i,j )) = 0 for all i < j is equivalent to all terms with the same degree vanishing. We denote its cubic terms as Q It is clear that Q n,n+1 = 0 if and only if ω n,n+2 = 1 for all n. We have which vanishes when ω n,n+3 = ω n−1,n+2 = 1 and ω n,n+1 = ω n+1,n+2 . We set ω n,n+1 = α.
Let k be the distance between i and j modulo M .
implying that ω i+1,j = ω i,j+1 = 1 for all i and j. This leads to ω i,j = 1 for all i and j. So far we have proved that ω n,n+1 = α for all n and ω i,j = 1 otherwise.
We are now ready to look at the rest terms in π I M (∂ t 1 (f i,j )). The condition π I M (∂ t 1 (f n,n+1 )) = 0 is equivalent to the following equation (we imply sums over r): In this expression, if we look at quadratic terms not containing u l , n−1 ≤ l ≤ n+2 as a factor, we get σ r n,n+1 = 0 if r / ∈ {n − 1, n, n + 1, n + 2}. We substitute them into (28) and get σ n−1 n,n+1 = σ n+2 n,n+1 = 0 after comparing to the quadratic terms in its both sides. We denote the sum over r of σ r n−1,n+1 u r by Σ n . The quadratic terms in (28) becomes which implies that Σ n is proportional to u n , and further leads to σ n+1 n,n+1 = σ n n,n+1 = Σ n = 0. Finally from the vanishing of linear terms in (28) we have η n,n+1 = η n,n+2 = 0. Thus we have that for all n, f n,n+1 = u n u n+1 − αu n+1 u n and f n,n+2 = u n u n+2 − u n+2 u n .
We will prove that f n,n+m = u n u n+m − u n+m u n for m > 2 by induction. Assume that we have for all 2 l k that f n,n+l = u n u n+l − u n+l u n . We now compute ∂ t 1 (f n,n+k ). Using the induction assumption we have 0 = π I M (∂ t 1 (f n,n+k )) = π I M (u n u n+k+1 u n+k − u n+k+1 u n+k u n + u n+k u n u n−1 − u n u n−1 u n+k ) = σ r n,n+k+1 u r u n+k − σ r n−1,n+k u n u r − η n−1,n+k u n + η n,n+k+1 u n+k .
Thus the coefficient σ r n,n+k+1 should be zero whenever r is not n but also whenever r is not n + k + 1 hence the σ's are identically zeros, from which it follows that η n,n+k+1 = 0. Hence we conclude the induction and complete the proof.
Note that the proof for the case M 5 can be directly generalized to the non-periodic case which means that the ideal I is the only stable ideal for the nonabelian Volterra flow within the class of ideals where f i,j has the form (23). This justifies our choice of the ideal I (4) in the case of infinite Volterra chain (2).

Bi-quantum structure of the periodic Volterra system with period 3
In the classical commutative case the M = 3 periodic Volterra system (22) is bi-Hamiltonian [24]. There are two compatible Poisson brackets defined by such that a linear combination of the Poisson brackets, called a Poisson pencil, is also a Poisson bracket for any choice of κ, i.e. the bracket {·, ·} κ is skew-symmetric and satisfies the Jacobi identity. The system admits two first integrals such that equations (22) with commutative variables can be written in a bi-Hamiltonian form These first integrals Poisson commute with each other and moreover, H 1 is in the kernel of the first Poisson bracket (is a Casimir element), while H 2 is in the kernel of the second one and According Proposition 4 and Theorem 5, the periodic Volterra system (22) on the free algebra A 3 admits a ∂ t 1 and † stable difference ideal I θ, = f (θ, ) n ; n ∈ Z 3 , generated by the polynomials depending on the two real parameters 0 ≤ < π, θ ∈ R. Thus, we have a pencil of quantised algebras A (θ, ) = A 3 I θ, . Algebra A (θ, ) has a central element where the self-adjoint elements are first integrals for the quantum Volterra system Moreover, system (33) in algebra A (θ, ) can be represented in the Heisenberg form With two quotient algebras A (θ,0) and A (0, ) we associate the following bi-quantum structure (a quantum deformation of the bi-Hamiltonian structure (30)) as follows: self-adjoint central element More work is required to study the quantum periodic Volterra systems with M ≥ 4 (25), (26) as we did for M = 3 above, which is not included in this paper.

Quantisation of periodic reductions of the cubic symmetry
In this section, we study the quantisation problem for periodical reductions of the cubic symmetry (3). In the infinite case this system admits two distinct quantisations (Proposition 2).
We claim that: 1. In the case M = 3 the quantisation ideal (23) is generated by relations (24).
3. For even M ≥ 6 there are two distinct quantisations corresponding to the ideal I a generated by the relations (26) and I b generated by relations The case M = 4 is exceptional, it admits three distinct quantisation ideals. One quantisation ideal is generated by by commutation relations (25) and the other two are generated by homogeneous quadratic commutation relations. The periodical reduction of the system (3) with the period M = 4 can be written in the form (Here we also add the constant q 2 following (19)): where the lower index n ∈ Z 4 . In the free algebra A 4 = C u 1 , . . . u 4 we consider the ideal I generated by six homogeneous quadratic polynomials f i,j , which depend on six nonzero constants ω i,j . The ideal I is ∂ t 2 -stable if and only if ∂ t 2 (f i,j ) ∈ I, 1 i < j 4. This is equivalent to the following system of equations on the parameters ω i,j ω 2 2,4 = 1, ω 2 1,4 ω 2 3,4 = 1, ω 2,3 = ω 2,4 ω 3,4 , ω 1,2 = ω 1,4 ω 2,4 ω 2 3,4 , ω 1,3 = ω 1,4 ω 3,4 .
Solving the above system of equations, we obtain the following statement: Theorem 6. A nonabelian system (35) admits a I-quantisation of the form (36) if and only if the six constants ω i,j take values as in one of four cases: and ω = q 2 = e 2i , where ∈ R. Moreover, in each of the above four cases the system (35) is a super-integrable quantum system. Proof. First note that the four cases listed in the statement correspond to the four solutions of the system (37). It is obvious that in each case the ideal is †-stable if and only if ω † = ω −1 . Thus we can set ω = e 2i , ∈ R.
We now prove the super-integrability of the obtained system in each case. Let which is a first integral for the quantum system (35) in all four cases. Moreover, in all four cases the quantum system (35) for self-adjoint variables u n can be written in the same Heisenberg form (21): In the case (a), corresponding to the quantisation of the Volterra system, the quantisation ideal I a is generated by the commutation relations between the variables u k as follows: The algebra A 4 I a has two central elements Since the central elements of the algebra commute with the Hamiltonian, they are first integrals of the system (38). The system of four equations (35) admits three commuting first integrals and therefore it is super-integrable.
In the case (b) the quantisation ideal I b is generated by the commutation relations between the variables u k as follows The dynamical system (35) on A 4 I b admits two first integrals The algebra A 4 I b has three central elements In the case (c), which is new, the quantisation ideal I c is generated by the commutation relations between the variables u k as follows: The dynamical system (35) on A 4 I c admits the first integral H 1 = u 3 u 1 commuting with H 2 . The algebra A 4 I c has two central elements The first integrals H 2 , H 1 and H 2 are obviously independent and therefore system (35) on A 4 I c is super-integrable.
The last case (d) can be obtained from the case (c) by the cyclic permutation of the variables {u 1 , u 2 , u 3 , u 4 } → {u 2 , u 3 , u 4 , u 1 }.
In the case M = 5 the only ∂ t 2 -stable ideal is defined by (26). The system admits three commuting first integrals where H is a central element of the algebra. The Heisenberg equations corresponding to H 1 and H 2 results in the periodic Volterra system and its cubic symmetry respectively.

Quantisation of the nonabelian Volterra Hierarchy
In this section, we extend Proposition 1 and Proposition 2 in Section 2 to the whole nonabelian Volterra hierarchy. We show that the quantum ideal I a (5) is invariant with respect to every member of the hierarchy (12) (Theorem 9) and that the quantum ideal I b (7) is invariant with respect to every even member of the nonabelian Volterra hierarchy that is, odd degree symmetries of the nonabelian Volterra equation (Theorem 14).
We are going to use the explicit expressions given by (12) to prove these statements. First we introduce some notations and definitions inspired by the monomials appearing in X (l) .
We introduce two sets of distinguished monomials, for k 1 We say that a k-degree monomial u α is admissible if α ∈ A k and is nonincreasing if α ∈ Z k .
Using these notations, we can simply write the expression X (k) given by (13) as Given an ideal I, either I a or I b , the canonical projection π I : A → A/I acts on X (k) as follows: where P I α (ω) is the unique polynomial in Z[ω] such that for α ∈ A k ∩ Z k , We often write it as P α (ω) if there is no ambiguity.
We say that two polynomials f, g ∈ A are I-equivalent denoted by f I ≃ g if f − g ∈ I. Polynomials f and g are I equivalent if and only if π I (f ) = π I (g).

Quantisation of the Volterra hierarchy
In this section, we will prove that the ideal I a defined by (5) is preserved by the symmetry flows (12), for all ℓ ∈ N.
We now assume that α ∈ A k and α − 1 ∈ A k . Consider the set E α defined as We split E α in two different ways by defining four subsets of E α : We are going to evaluate each term in it. Note that A α = A k is the set of all elements equivalent to α. Thus by definition (43), we have For any β ∈ B α , we have β − 1 ∈ A k and β − 1 ∼ α − 1 and thus Let β ∈ D α . There is β i > 0 for some 0 < i < k since β ∼ α and α − 1 ∈ A k . Assume that there are 0 < m k positive components at positions i 1 i 2 · · · i m in β. Starting from i 1 , we find the first zero entry on the left of i 1 , that is, l 1 = max 1 j i 1 −1 {β j = 0} and move the components from l 1 to i 1 − 1 to the right of i 1 and obtain β 1 with For β 1 , we find the first zero entry on the left of i 2 , that is, l 2 = max l 1 +1 j i 2 −1 β 1 j = 0 and move the components from l 2 to i 2 − 1 to the right of i 2 and obtain β 2 . We repeat this procedure for all positive components in β. Thus we obtain a k-component vector γ = β l ∈ A α . This leads to Similarly, let β ∈ C α . There is β i 0 for some 0 < i < k since β ∼ α and α ∈ A k . For all nonpositive components, we move the first component being 1 on its right to its left, taking with all the components of β on its left that are larger than 1. Thus we obtain a k-component vector γ ∈ B α . This leads to We substitute (46)-(49) into (45) and thus we obtain the required identity (44).
Proof. We fix k and let u τ = Q (k) be the (k + 1)-degree symmetry of the Volterra equation given by (12). Since S(I) = I we only need to show that This means that We rewrite it in terms of X. Here we simply drop its upper index of X (k) .
It is clear that, for any α ∈ Z k , we have Note that for all l ∈ Z, we have Here the sum is over all α ∈ Z k including the ones not in A k . Hence, the left-handed side of (52) becomes For any α ∈ Z k , we need to check that the coefficient of π Ia (u α uu m ) vanishes. Using Corollary 8, it amounts to compute which equals zero after the simplification and thus we complete the proof.

Non-deformation quantisation for all odd-degree Volterra symmetries
In this section, we will prove that all odd-degree symmetries of the nonabelian Volterra hierarchy admit the quantisation I b , that is, the ideal I b defined by (7) is preserved by the symmetry flows (12) when ℓ is even. We extend the automorphism S and the antiautomorphism T to the algebra A[ω] by letting S(ω) = T (ω) = −ω so that these operators are well-defined on the quotient A/I b .
The ideas guiding the proof essentially are the same as in the previous section with the notable difference of the equivalence of Proposition 7, which is much harder in this case.
To prove that the ideal I b defined by (7) is preserved by the symmetry flows Q (2k) , we first prove the equivalents of Proposition 7 only in this case for α ∈ Z 2k . We now assume that α ∈ A 2k and α − 1 ∈ A 2k . In the same way as we prove Proposition 7, we define the set E α as and split E in two different ways by defining four subsets of E α : It follows that We need to evaluate each term under the ideal I b . Since A α = A 2k is the set of all elements equivalent to α, it follows from (43) that For any β ∈ B, note that β − 1 ∈ A 2k and β − 1 ∼ α − 1 and thus We are now left to evaluate the terms for D α and for C α and we do so in Proposition 10 and Proposition 11, respectively.
Thus in the quotient algebra, we obtain We denote n i=1 |a i ||b i | by a · b and note that ν(µ, 1) = ν(α, 1) and ν(µ, 2) = ν(α, 2). Hence Note that the first term gives us the required identity (56) using (54). Thus we are left to prove that From now on, we identify a pair of vectors ( a, b) with n i=1 u a i uu b i . Let We split this set in two equal parts Y and Z after the following remarks. Let c be the number of indices i such that |a i | and |b i | are both odd, d the number of indices such that |a i | and |b i | are both even. When none of this is true, the parity of |a i | + |b i | is odd.
Since the length of α is even , the parity of |µ| + |γ| is the same as n. Hence, which implies that c + d is even. Moreover, we know that a · b is odd, that is, Thus we have that both c and d are odd.
Let I = {i 1 , ..., i c+d } be the set of indices i such that |a i | + |b i | is even (We know that this set has cardinal c + d). Let l be minimal so that |a i l | and |a i c+d+1−l | have different parity. Such l exists and is unique. Indeed, if it did not exist we would have |a i l | ≡ |a i l+c+d−1 | for all l implying that c and d are even.
We denote i l by k( a, b) and i c+d+l−1 by m( a, b). However, in the sequel we will abuse notation and simply write k and m, knowing that we have fixed the element ( a, b) in the set Σ. Based on these definitions, we put the pair ( a, b) in the set Y if |a k | is odd and we put it in Z if |a k | is even.
Let q ∈ Y and u q = n i=1 u a i uu b i . We are going to construct a bijective map φ : Y → Z such that φ(u q ) where the maps ξ i are defined in Lemma 17 in Appendix. Thus φ only transforms the product from the block k to the block m, i.e., m i=k u a i uu b i . By definition of the maps ξ i , if we represent φ(u q ) as ( c, d) we see that c k and d k will have even length and that c m and d m will have odd length. It means that φ(u q ) is an element of Z, but also that we still have k(φ(u q )) = k and m(φ(u q )) = m. That is because we have left the first k − 1 blocks and the last n − m blocks intact. Since the values of k and m are unchanged by φ and that all the ξ i 's are bijections, it follows that φ is a bijection as well. So it only remains to check that We know that |b k | = 1 mod 2 and |a m | = 0 mod 2. Hence, (|a i | + |b i | + 1) mod 2 = 1 mod 2 since there is a even number of indices i for which |a i | ≡ |b i | between k and m.
Below we give an example to illustrate this proposition.
Following the line of Proposition 10's proof, with this example we first give a full description of the set Σ, then split it as Σ = Y ∪ Z. An admissible monomial is given by a partition of |a 1 | + |a 2 | + |a 3 | = 2 and a partition |b 1 | + |b 2 | + |b 3 | = 1. For this monomial to be in Σ we need |a 1 ||b 1 | + |a 2 ||b 2 | + |a 3 ||b 3 | to be odd. It must be that (|b 1 |, |b 2 |, |b 3 |) is one of (1, 0, 0), (0, 1, 0) and ( For each element in Y , we first identify the blocks k and m, to remove a 1 and a −1 from the block k and to add them to the block m. We now write Z in the same order, that is, Z = φ(Y ): One can check that π I b ( β∈Σ u β ) = 0 and π I b ( β∈Y u β ) = −π I b ( β∈Z u β ).
Having evaluated all terms in (53), we are now in the position to prove the similar result as Proposition 7 for the ideal I b defined by (7).
Similar to Corollary 8 for the case of ideal I a , we have the following statement for the case of ideal I b : Corollary 13. Let α ∈ Z 2k . There exists a non zero rational function R α (ω) ∈ Q(ω) such that Proof. Without the loss of generality, we assume that α − l ∈ A 2k , for 0 l q. Let The identity (58) implies that Thus for 0 l q we have When α + m / ∈ A 2k , we have P α+m (ω) = 0 following the definition of (43).
Theorem 14. The quantisation ideal I b is stable with respect to every even member of the Volterra is the sum of all admissible monomials of size 2ℓ, ℓ 1. Let k 2. We want to show that ∂ τ (uu k + u k u) is in the ideal I b . By definition of u τ , this means that or, in terms of X (we drop its upper index): Let us fix an element β ∈ Z 2ℓ . We are going to show that the terms equivalent to u β uu k modulo multiplication by an element of Z[ω] in (61) cancel out. It is clear that We know that for all m ∈ Z, We compute the terms coming from X −k−1 , X 1 and X −1 in a similar way. Thus, to prove that the coefficient of u β uu k in (61) is zero amounts to check that Using Corollary 13, we need to verify and thus the identity (60) holds. The proof that π I b ∂ τ (u k u k+1 − (−1) k ωu k+1 u k ) = 0 for all k ∈ Z is similar and we will not repeat it.

Summary and discussion
In this paper we develop the method of quantisation of dynamical systems defined on free associative algebras based on the concept of quantisation ideals [5]. It enables us to determine possible commutation relations between the dynamical variables which are consistent with the dynamical system and define associative multiplication in the quotient algebra. The method does not use any information on the Poisson structure of the dynamical system and enables us to find non-deformation quantisations of the system. To determine commutation relations consistent with a system is a very first step to its quantum theory. Next steps will require the development of the representation theory for the quantised algebras obtained and study the spectral theory of the operators involved.
In this paper we explicitly proved that the nonabelian Volterra system (2) and its infinite hierarchy of symmetries admit the deformation quantisation with commutation relations (6). We also proved that the sub-hierarchy, consisting of all odd degree symmetries, admits a non-deformation quantisation with commutation relations (8). The existence of non-deformation quantisations is quite surprising. Further study is required to explore the properties of these new remarkable quantum algebra and quantum integrable equations.
Recently, when the paper has already been submitted to the journal, we found explicit expressions for the infinite sequence of quantum Hamiltonians H n corresponding to the I a quantisation of the Volterra hierarchy where A ℓ 0 = {α ∈ A ℓ ∩ Z ℓ ; α ℓ = 0}. Assuming that ω = e 2i , ∈ R, the Hamiltonians H ℓ are self-adjoint H † ℓ = H ℓ . They commute with each other, and the dynamical equations of the quantum hierarchy can be written in the Heisenberg form (compare with (20)): We have also found explicit expressions for self-adjoint commuting quantum Hamiltonians corresponding to non-deformation quantisation (8) and present the quantum hierarchy with even times in the Heisenberg form. A detail proof of these results will be published elsewhere soon.
The Volterra hierarchy admits periodic reductions with any positive integer period M . We have shown that the Volterra system with periods M = 3, 4 admit quantisations with non-homogeneous commutation relations (Theorem 5). When M = 3, we proved the resulting quantum system is not only super integrable but also admits bi-quantum structure, similar to its bi-Hamiltonian structure in the classical case. The cubic symmetry of the Volterra system with period M = 4 admits three distinct quantisations. In each case, the quantum system is a super-integrable systems (Theorem 6). Systems with periods M ≥ 5 require more work, they have not been studied in this paper in any detail.
The methods developed in [5] and this paper can be applied to the nonabelian Narita-Itoh-Bogoyavlensky lattice [17] The Volterra equation is corresponding to the case when p = 1. Our study shows that system (62) and all equations of its hierarchy admit the quantisation with commutation relations u n u n+k = ωu n+k u n , 1 k p, u n u m = u m u n , |n − m| > p n, m ∈ Z, where ω is a nonzero constant. The proof of this statement will be published elsewhere. These commutation relations were also obtained by Inoue and Hikami [20] using ultra-local Lax representation and R-matrix technique.
Besides quadratic ideals, our computations for the nonabelian Volterra equation and its lower degree symmetries suggest that there is a ∂ t ℓ -stable ideal generated by quadratic and cubic homogeneous polynomials. For example, as far as we have checked, the first few symmetries in the nonabelian Volterra hierarchy leave the following cubic ideal invariant: Further research is needed to study the properties of the Volterra chain which is well defined on the quotient algebra A Ĩ . Very little is known about this new invariant ideal and the quotient algebra which does not satisfy the condition (ii).
The concept of quantisation ideals has not been linked yet with Lax representations, recursion operators, master-symmetries and other objects associated with the theory of integrable systems. We think that further development of this theory will enable us to embrace a wide range of integrable systems as well as to clarify and simplify rather technical proofs of the statements presented in this paper. Appendix: Lemmas used for the proof of Proposition 10

Acknowledgments
In this appendix, we are going to prove the lemmas used in constructing the bijection map between sets A α and D α (Proposition 10) in Section 5.2.
Let l be any integer. We denote by Λ l the set of admissible monomials of the form u a u l u b satisfying (i) both a and b have components greater than l if they are not empty.
(ii) there exists a suffix d of a of odd length a = cd where c is either empty or ends with l + 1.
(iii) if b is non-empty then it ends with l + 1.
If the length of d in (ii) is minimal, we say that d is the minimal odd suffix of a.
We denote by Γ l the set of admissible monomials of the form u a u l u b where (i) both a and b have components greater than l.
(ii) there exists a prefix c of b of odd length b = cd where c ends with l + 1.
If the length of c in (ii) is minimal, we say that c is the minimal odd prefix of b.
Lemma 15. For all l ∈ Z, we construct a bijection ψ : Λ l → Γ l such that for all x ∈ Λ l , π I b (ψ(x)) = (−1) l ω x. Moreover, if x = u a u l u b and ψ(x) = u c u l u d , then |c| = |a| − |m| and |d| = |b| + |m|, where m is the minimal odd suffix of a.
Proof. We construct ψ by induction on |a| + |b|. The only element of length 2 in Λ l is u l+1 u l , while the only element of length 2 in Γ l is u l u l+1 . We let ψ(u l+1 u l ) = u l u l+1 . The minimal odd suffix of u l+1 is itself and we have π I b (u l u l+1 ) = (−1) l ωu l+1 u l , hence the statement of the Lemma holds for elements of length 2.
Suppose that we have constructed ψ for all lengths strictly less than n satisfying the statement. We now construct ψ for elements of length n and prove it satisfies the statement. Let u a u l u b be an element of Λ l of length n. Let d be the minimal odd suffix of a. Explicitly, this u d has the form u e u l+1 u d 1 u l+1 ...u dp u l+1 , where the |d i |'s are odd and |e| is even (hence possibly e is empty). Note that in this decomposition of u d , the elements d i and e do not contain any j < l + 2 and all end with l + 2 (except if e is empty). Hence for all i = 1, ..., p, u l+1 u d i is an element of Γ l+1 whose length is strictly less than n. By the induction hypothesis, there exist f i of odd length and g i of even length such that ψ −1 (u l+1 d i ) = u f i u l+1 u g i .
Note that f i does not have a proper odd suffix due to the last assertion in the Lemma. Recall that all elements in f i and g i are greater than l + 1. The element u e ψ −1 (u l+1 u d 1 )...ψ −1 (u l+1 u dp )u l+1 is well-defined. It has exactly the same (odd) length as d without any proper odd prefix and π I b (u e ψ −1 (u l+1 u d 1 )...ψ −1 (u l+1 u dp )u l+1 ) = ((−1) l+1 ω) −p u e u l+1 u d 1 u l+1 ...u dp u l+1 .
We know that there are as many elements of length n in Γ l as in Λ l , hence it remains to check the injectivity of ψ for length n. Suppose that we have ψ(u a u l u b ) = ψ(uãu l ub). In other words, we have u c u l u e ψ −1 (u l+1 u d 1 )...ψ −1 (u l+1 u dp )u l+1 u b = ucu l uẽψ −1 (u l+1 ud 1 )...ψ −1 (u l+1 ud q )u l+1 ub This equality implies that c =c so we can simplify it slightly: u e ψ −1 (u l+1 u d 1 )...ψ −1 (u l+1 u dp )u l+1 u b = uẽψ −1 (u l+1 ud 1 )...ψ −1 (u l+1 ud q )u l+1 ub Recall that u e ψ −1 (u l+1 u d 1 )...ψ −1 (u l+1 u dp )u l+1 is the minimal odd prefix of the left hand side and that uẽψ −1 (u l+1 ud 1 )...ψ −1 (u l+1 ud q )u l+1 is the minimal odd prefix of the right hand side. By unicity of the minimal odd prefix, they are equal. In particular, we have b =b and p = q. Recall the definition of f i and g i such that ψ −1 (u l+1 u d i ) = u f i u l+1 u g i . Similarly we write ψ −1 (u l+1 ud i ) = uf i u l+1 ug i . We have u g 0 u f 1 u l+1 u g 1 u − f 2 u l+1 ...u fp u l+1 u gp = ug 0 uf 1 u l+1 ug 1 uf 2 u l+1 ...uf p u l+1 ug p , where we have let g 0 = e andg 0 =ẽ. Therefore we have for all i = 0, ..., p − 1 Recall that both f i+1 andf i+1 are their own minimal odd suffix. Hence f i+1 is the minimal odd suffix of g i f i+1 andf i+1 is the minimal odd suffix ofg ifi+1 . By unicity of the minimal odd suffix we have f i+1 =f i+1 , from where it follows that g i =g i . Hence u l+1 u d i = ψ(u f i u l+1 u g i ) = ψ(uf i u l+1 ug i ) = u l+1 ud i and thus we complete the proof.
Let l be any integer. We denote by Θ l the set of admissible monomials of the form u a u l u b where (i) both a and b have components strictly smaller than l.
(ii) there exists a suffix d of a of odd length a = cd where d starts with l − 1.
If the length of d in (ii) is minimal, we say that d is the minimal odd suffix of a.
We denote by Φ l the set of admissible monomials of the form u a u l u b where (i) both a and b have components strictly smaller than l.
(ii) there exists a prefix c of b of odd length b = cd where d is either empty or starts with l − 1.
(iii) a is either empty or starts with l − 1.
If the length of c in (ii) is minimal, we say that c is the minimal odd prefix of b.
Lemma 16. For all l ∈ Z, we construct a bijection ρ : Θ l → Φ l such that π I b (ρ(x)) = (−1) l+1 ω −1 x for all x ∈ Θ l . Moreover, if x = u a u l u b and ψ(x) = u c u l u d , then |c| = |a| − |m| and |d| = |b| + |m|, where m is the minimal odd suffix of a.
Let m be the minimal odd prefix of T (a). We know that ψ −1 (T (b)u −l T (a)) = cu −l d with |c| = |T (b)| + |m| and |d| = |T (a)| − |m|. We have ρ(au l b) = T (d)u l T (c). We conclude by noting that T (m) is the minimal odd suffix of a.
Recall that we identify an element of Σ, that is a pair ( a, b) such that a · b = 1 mod 2 with the product n i=1 u a i uu b i . We denote a subset of X consisting of a part of Σ such that u a j u ∈ Λ 0 and u b j u ∈ Θ 0 for some 1 j n by Σ j . We are going to construct bijections ξ j : Σ j → Σ j+1 . Lemma 17. There exists a bijection ξ j : Σ j → Σ j+1 , 1 j n − 1, so that ξ j (u p ) Proof. Let ( a, b) be an element of Σ j . Consider the product of block j with block j + 1, i.e., u a j uu b j u a j+1 uu b j+1 .
We now define ξ j : (( a, b) → ( c, d) as follows: c i = a i and d i = b i if i = j and i = j + 1 c j =ã j , d j =b j , c j+1 =ã j , and d j+1 =b j .

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