The algebra of q-pseudodifferential symbols and the q-WKP(n) algebra (original) (raw)
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Extensions and contractions of the Lie algebra of q-pseudodifferential symbols
1994
We construct cocycles on the Lie algebra of pseudo- and q-pseudodifferential symbols of one variable and on their close relatives: the sine-algebra and the Poisson algebra on two-torus. A ``quantum'' Godbillon-Vey cocycle on (pseudo)-differential operators appears in this construction as a natural generalization of the Gelfand-Fuchs 3-cocycle on periodic vector fields. We describe a nontrivial embedding of the Virasoro algebra
Communications in Mathematical Physics, 1993
The KP hierarchy is hamiltonian relative to a one-parameter family of Poisson structures obtained from a generalized Adler map in the space of formal pseudodifferential symbols with noninteger powers. The resulting W-algebra is a one-parameter deformation of WKp admitting a central extension for generic values of the parameter, reducing naturally to W, for special values of the parameter, and contracting to the centrally extended W1 +o~, Woo and further truncations. In the classical limit, all algebras in the one-parameter family are equivalent and isomorphic to w~:p. The reduction induced by setting the spin-one field to zero yields a one-parameter deformation of V~/~o which contracts to a new nonlinear algebra of the W~-type.
q-deformed W-algebras and elliptic algebras
1998
The elliptic algebra A q,p (sl(N) c) at the critical level c = −N has an extended center containing trace-like operators t(z). Families of Poisson structures, defining q-deformations of the W N algebra, are constructed. The operators t(z) also close an exchange algebra when (−p 1/2) N M = q −c−N for M ∈ Z. It becomes Abelian when in addition p = q N h where h is a non-zero integer. The Poisson structures obtained in these classical limits contain different q-deformed W N algebras depending on the parity of h, characterizing the exchange structures at p = q N h as new W q,p (sl(N)) algebras.
Models of q‐algebra representations: q‐integral transforms and ‘‘addition theorems’’
Journal of Mathematical Physics, 1994
In his classic book on group representations and special functions Vilenkin studied the matrix elements of irreducible representations of the Euclidean and oscillator Lie algebras with respect to countable bases of eigenfunctions of the Cartan subalgebras, and he computed the summation identities for Bessel functions and Laguerre polynomials associated with the addition theorems for these matrix elements. He also studied matrix elements of the pseudo-Euclidean and pseudo-oscillator algebras with respect to the continuum bases of generalized eigenfunctions of the Cartan subalgebras of these Lie algebras and this resulted in realizations of the addition theorems for the matrix elements as integral transform identities for Bessel functions and for confluent hypergeometric functions. Here we work out q analogs of these results in which the usual exponential function mapping from the Lie algebra to the Lie group is replaced by the q-exponential mappings Eq and eq. This study of representations of the Euclidean quantum algebra and the q-oscillator algebra (not a quantum algebra) leads to summation, integral transform, and q-integral transform identities for q analogs of the Bessel and confluent hypergeometric functions, extending the results of Vilenkin for the q=1 case.
The (N,M)-th KdV hierarchy and the associated W algebra
We discuss a differential integrable hierarchy, which we call the (N, M)-th KdV hierarchy, whose Lax operator is obtained by properly adding M pseudo-differential terms to the Lax operator of the N-th KdV hierarchy. This new hierarchy contains both the higher KdV hierarchy and multifield representation of KP hierarchy as subsystems and naturally appears in multi-matrix models. The N + 2M − 1 coordinates or fields of this hierarchy satisfy two algebras of compatible Poisson brackets which are local and polynomial. Each Poisson structure generate an extended W 1+∞ and W ∞ algebra, respectively. We call W (N, M) the generating algebra of the extended W ∞ algebra. This algebra, which corresponds with the second Poisson structure, shares many features of the usual W N algebra. We show that there exist M distinct reductions of the (N, M)-th KdV hierarchy, which are obtained by imposing suitable second class constraints. The most drastic reduction corresponds to the (N + M)-th KdV hierarchy. Correspondingly the W (N, M) algebra is reduced to the W N +M algebra. We study in detail the dispersionless limit of this hierarchy and the relevant reductions. * This integrable structure also shows up in WZW model and Conformal Affine Toda field theories (CAT models)[6]. For a more mathematical approach see [7] and references therein * The product is symmetric since < AB >=< BA >, while invariance means < A[B, C] >=< [A, B]C >. Occasionally we will also denote the product in other ways: < AB >= A(B) = T r(AB). † The usual adjoint action is Ad Y X = [X, Y ], here we use the same notation to denote the adjoint action generated by the R-commutator.
Hamiltonian reduction and the construction of q -deformed extensions of the Virasoro algebra
Journal of Physics A: Mathematical and General, 1998
In this paper we employ the construction of Dirac bracket for the remaining current of sl(2) q deformed Kac-Moody algebra when constraints similar to those connecting the sl(2)-WZW model and the Liouville theory are imposed and show that it satisfy the q-Virasoro algebra proposed by Frenkel and Reshetikhin. The crucial assumption considered in our calculation is the existence of a classical Poisson bracket algebra induced, in a consistent manner by the correspondence principle, mapping the quantum generators into commuting objects of classical nature preserving their algebra. 1 Supported by FAPESP 2 Work partially supported by CNPq 3 Supported by CNPq
The algebra of differential operators on the circle and W KP (q)
Letters in Mathematical Physics, 1993
Radul has recently introduced a map from the Lie algebra of differential operators on the circle to W n. In this note we extend this map to W (q) KP , a recently introduced one-parameter deformation of W KP-the second hamiltonian structure of the KP hierarchy. We use this to give a short proof that W ∞ is the algebra of additional symmetries of the KP equation.
The (N,M)th Korteweg–de Vries hierarchy and the associated W-algebra
Journal of Mathematical Physics, 1994
We discuss a differential integrable hierarchy, which we call the (N, M )-th KdV hierarchy, whose Lax operator is obtained by properly adding M pseudo-differential terms to the Lax operator of the N -th KdV hierarchy. This new hierarchy contains both the higher KdV hierarchy and multifield representation of KP hierarchy as sub-systems and naturally appears in multi-matrix models. The N + 2M − 1 coordinates or fields of this hierarchy satisfy two algebras of compatible Poisson brackets which are local and polynomial. Each Poisson structure generate an extended W 1+∞ and W ∞ algebra, respectively. We call W (N, M ) the generating algebra of the extended W ∞ algebra. This algebra, which corresponds with the second Poisson structure, shares many features of the usual W N algebra. We show that there exist M distinct reductions of the (N, M )-th KdV hierarchy, which are obtained by imposing suitable second class constraints. The most drastic reduction corresponds to the (N + M )-th KdV hierarchy. Correspondingly the W (N, M ) algebra is reduced to the W N +M algebra. We study in detail the dispersionless limit of this hierarchy and the relevant reductions. * This integrable structure also shows up in WZW model and Conformal Affine Toda field theories (CAT models) . For a more mathematical approach see and references therein * The product is symmetric since < AB >=< BA >, while invariance means < A[B, C] >=< [A, B]C >. Occasionally we will also denote the product in other ways: < AB >= A(B) = T r(AB).
Realizations of q-Deformed Virasoro Algebra
Progress of Theoretical Physics, 1993
We investigate the q-deformed Virasoro algebra presented by Curtright and Zachos. After showing some new results on the central extension and the operator product expansion, we discuss the relation between the q-deformed Virasoro algebra and the Volterra Poisson bracket algebra. The realization in terms of an infinite set of oscillators is also discussed from the viewpoint of a deformation of the Poisson bracket.
Brackets with (τ,σ)-derivations and (p,q)-deformations of Witt and Virasoro algebras
Forum Mathematicum, 2015
The aim of this paper is to study some brackets defined on (τ, σ)-derivations satisfying quasi-Lie identities. Moreover, we provide examples of (p, q)-deformations of Witt and Virasoro algebras as well as sl(2) algebra. These constructions generalize the results obtained by Hartwig, Larsson and Silvestrov on σ-derivations, arising in connection with discretizations and deformations of algebras of vector fields.