Quantisation ideals of nonabelian integrable systems
aa r X i v : . [ n li n . S I] S e p Quantisation ideals of nonabelian integrable systems
A.V.Mikhailov
We consider dynamical systems on the space of functions taking values in a free associative algebra[1, 2]. The system is said to be integrable if it possesses an infinite dimensional Lie algebra of commutingsymmetries. In this paper we propose a new approach to the problem of quantisation of dynamical systems,introduce the concept of quantisation ideals and provide meaningful examples.In order to illustrate the new approach we have chosen the following nonabelian integrable systems: theVolterra chain (i) and the Bogoyavlensky N –chains (ii) [3] (i) u t = u u − uu − , (ii) dudt = N X k =1 ( u k u − uu − k ) , (1)These are infinite systems of equations where we use standard notations u = u = u ( n, t ) , u k = u k ( n, t ) = u ( n + k, t ) , n, k ∈ Z . In equations (1) functions u k are elements of a free associative algebra A = K h . . . u − , u, u , . . . i over a zero characteristic field of constants K .Traditional quantisation theories start with classical systems on functions taking values in commutativealgebras. In this paper we propose a new approach departing from the systems defined on free associativealgebras. In this approach the quantisation problem is reduced to a description of two-sided ideals whichdefine the commutation relations in the quotient algebras and are invariant with respect to the dynamicsof the system. We begin with consideration of two-sided ideals J F ⊂ A generated by an infinite set ofpolynomials of the form J F = h{ F p,q = u q u p − ω p,q u p u q | p, q ∈ Z , p > q, ω p,q ∈ K ∗ }i (2)and find out such structure constants ω p,q , that equations (1) could be restricted to the quotient algebra A J F = A (cid:30) J F . In such a case we say that system (1) admits J F –quantisation and is defined on the quantisedalgebra A J F .In general, by quantised algebra it is understood such a non-commutative quotient algebra A J = A (cid:30) J thathas an additive basis of lexicographically ordered monomials of the form u a i u a i · · · u a n i n in finite or infinitenumber of variables, where i < i < · · · < i n , i k ∈ Z . The ideal J defines commutation relations in thequotient algebra and we call it the quantisation ideal . In the case of finitely generated algebras it is said that A J has a Poincar´e–Birkhoff-Witt basis.Algebra A has a natural automorphism generated by the shift operator S ( u k ) = u k +1 , k ∈ Z . Derivations D f of algebra A , commuting with the automorphism S , are called evolutionary. It is sufficient to define anevolutionary derivation on one algebra generator D f ( u ) = f ∈ A . The evolutionary derivation D f ( u ) is inone-to-one correspondence with the system of differential-difference equations du k dt f = S k f f = f ( . . . , u − , u, u , . . . ) ∈ A , k ∈ Z . (3)By a symmetry of equation (3) we understand an evolutionary derivation D g which commutes with D f , or,in other words, the evolutionary differential-difference system of equations u t g = g , which is compatible with(3). Symmetries D g , D g are called commuting, if [ D g , D g ] = 0 . A derivation D f of the algebra A , whichmaps the ideal J ⊂ A into itself D f ( J ) ⊆ J , induces canonically a derivation D f on A J . If S ℓ ( J ) ⊆ J for acertain integer number ℓ > , then S ℓ is automorphism of A J commuting with D f . It enables us to restrictsystem (3) to A J .Let us begin with non-abelian Volterra system (1) (i) and its symmetry dudτ = uu − u − + uu − u − + uuu − − u uu − u u u − u u u. (4)1 roposition 1 Equation (4) can be restricted to A J F only in the following cases: (i) ω n +1 ,n = α, ω n,m = 1 , (ii) ω n +1 ,n = ( − n α, ω n,m = − n − m ≥ . Nonabelian Volterra system can be restricted to the algebra A J F only in the case (i) . There are reasons to state that all symmetries of the Volterra hierarchy admit quantisation (i) for which S ( J F ) = J F . If we restrict ourself with odd-degree equations from the hierarchy, then there exists analternative quantisation (ii) for which S ( J F ) = J F . Proposition 2
Nonabelian N –chain (1) (ii) admits J F –quantisation only in the case ω n + k,n = α, where ≤ k ≤ N, α = 0 , and ω n,m = 1 , for n − m > N. Proposition 3
There exists a modification u t = u u uu + u uu − u − uu uu − − uuu − u − (5) of the nonabelian N = 2 Bogoyavlensky chain. System (5) admits J F –quantisation only in the case ω n +1 , m = α, ω n +2 , m = β, ω n +3 , m = α − β − , α, β = 0 , n ≥ m, n, m ∈ Z . The quantisation of the chain (5) is defined by two independent “deformation” parameters α, β . For biggervalues of N the number of independent parameters may also increase. In the case N = 3 a quantisation ofthe corresponding modified chain is defined by four independent parameters.Periodic closures of the chains u k + M = u k with period M result in nonabelian systems of ordinarydifferential equations for functions taking their values in a finitely generated unital free algebra A M = K h u , . . . , u M i . In these cases the system may admit inhomogeneous quantisation ideals of the form J G = h{ G p,q = u q u p − ω p,q u p u q + M X r =1 σ rp,q u r + η p,q | p, q ∈ Z , p > q, ω p,q = 0 , ω p,q , σ rp,q , η p,q ∈ K }i Proposition 4
Nonabelian periodical Volterra chain (1) (i) with period M admits J G –quantisation iff thefollowing commutation relations M = 2 : uu = αu u + βu + γu + η ; M = 3 : u n u n +1 = αu n +1 u n + β ( u + u + u ) + η, n ∈ Z ; M = 4 : u u = αu u + βu + γu − βγ, u u = u u − βu + βu ,u u = αu u + βu + γu − βγ, u u = αu u + βu + γu − βγ,u u = u u − γu + γu , u u = αu u + βu + γu − βγ ; M ≥ u n +1 u n = αu n u n +1 , u n u m = u m u n , | n − m | > , n, m ∈ Z M . take place. The constants α, β, γ, η ∈ K , α = 0 are arbitrary. The author is grateful to V.M.Buchstaber and V.V.Sokolov for useful discussions, and the EPSRC grantEP/P012655/1 for partial support.
Список литературы [1] P. Etingof, I. Gelfand, and V. Retakh
Nonabelian Integrable Systems, Quasideterminants, AndMarchenko Lemma.
Mathematical Research Letters 5, 1–12, 1998[2] A.V. Mikhailov and V.V. Sokolov
Integrable ODEs on associative algebras , CMP 211(1):231–251, 2000[3] O. I. Bogoyavlenskii