Quasi-Linear Algebras and Integrability (the Heisenberg Picture)

Abstract

Abstract. We study Poisson and operator algebras with the "quasi-linear property" from the Heisenberg picture point of view. This means that there exists a set of one-parameter groups yielding an explicit expression of dynamical variables (operators) as functions of "time" t. We show that many algebras with nonlinear commutation relations such as the Askey-Wilson, q-Dolan-Grady and others satisfy this property. This provides one more (explicit Heisenberg evolution) interpretation of the corresponding integrable systems. Classical versionAssume that there is a classical Poisson manifold with the Poisson brackets (PB) {x, y} defined for all dynamical variables x, y belonging to this manifold. Of course, it is assumed that the PB satisfy the standard conditions: (i) {x, α 1 y + α 2 z} = α 1 {x, y} + α 2 {x, z} -linearity (α 1 , α 2 are arbitrary constants);(ii) {x, y} = −{y, x} -antisymmetricity; (iii) {x, yz} = z{x, y} + y{x, z} -the Leibnitz rule; (iv) {{x, y}, z} + {{y, z}, x} + {{z, x}, y} = 0 -the Jacobi identity.If one chooses the "Hamiltonian" H (i.e. some dynamical variable belonging to this manifold), then we have the standard Hamiltonian dynamics: all variables x(t) become depending on an additional time variable t and the equation of motion is defined aṡ x(t) = {x(t), H}.(1.1) Of course, the Hamiltonian H is independent of t, becauseḢ = {H, H} = 0. More generally, a dynamical variable Q is called the integral of motion ifQ = 0. Clearly, this is equivalent to the condition {Q, H} = 0. From (1.1) we haveẍ = {{x, H}, H} and more generally d n x dt n = {. . . {x, H}, H}, . . . H} = {x, H (n) }, This paper is a contribution to the

The Universal Askey-Wilson Algebra

“…The Z 3 -symmetric quantum algebra O ′ q (so 3 ) [18, Remark 6.11], [22], [23,Section 3], [28,29,35,48] is a special case of AW, and the recently introduced Calabi-Yau algebras [21] give a generalization of AW. The algebra AW plays a role in integrable systems [2,7,8,9,10,11,12,13,14,15,41,42,43,61] and quantum mechanics [46,47], as well as the theory of quadratic algebras [35,36,49]. There is a classical version of AW that has a Poisson algebra structure [25], [40, equation (2.9)], [45, equations (26)- (28)], [66].…”

Finite-Dimensional Irreducible Modules of the Universal Askey–Wilson Algebra

“…Hence (31) is surjective. Now, applying the rank-nullity theorem to (31) the equality (27) follows. Lemma 6.6.…”

A classification of sharp tridiagonal pairs

“…Some notable papers on the topic are [6, 13, 28-31, 35-37, 71]. There are connections to representation theory [2,9,21,26,29,31,39,41,42,64,65,75], partially ordered sets [73], the bispectral problem [5,6,[22][23][24]82], statistical mechanical models [8-14, 17-20, 62], and other areas of physics [61,81,83].…”