The algebra of differential operators on the circle and W KP (q) (original) (raw)
Related papers
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.
A Note on Symmetries and Generalized W∞ Algebra of the Modified KP Equation
Letters in Mathematical Physics, 1997
Some remarks on the paper 'Symmetries and generalized W1 algebra of the modified KP equation' (S. Y. Lou and G. J. Ni, Lett. Math. Phys. 34 (1995), 327-331) are given. It is pointed out that if we consider the inverse operator of a differential operator to be a linear operator, the vector fields v n f defined in the above Letter are not certain to be symmetries of the modified KP equation under consideration.
The algebra of q-pseudodifferential symbols and the q-WKP(n) algebra
Journal of Mathematical Physics, 1996
In this paper we continue with the program to explore the topography of the space of Wtype algebras. In the present case, the starting point is the work of Khesin, Lyubashenko and Roger on the algebra of q-deformed pseudodifferential symbols and their associated integrable hierarchies. The analysis goes on by studying the associated hamiltonian structures for which compact expressions are found. The fundamental Poisson brackets yield q-deformations of W KP and related W-type algebras which, in specific cases, coincide with the ones constructed by Frenkel and Reshetikhin. The construction underlies a continuous correspondence between the hamiltonian structures of the Toda lattice and the KP hierarchies.
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).
1992
In this paper we study the inter-relationship between the integrable KP hierarchy, nonlinear hatWinfty\hat{W}_{\infty}hatWinfty algebra and conformal noncompact SL(2,R)/U(1)SL(2,R)/U(1)SL(2,R)/U(1) coset model at the classical level. We first derive explicitly the Possion brackets of the second Hamiltonian structure of the KP hierarchy, then use it to define the hatW1+infty\hat{W}_{1+\infty}hatW1+infty algebra and its reduction hatWinfty\hat{W}_{\infty}hatWinfty. Then we show that the latter is
Bihamiltonian structure of the KP hierarchy and the WKP algebra
Physics Letters B - PHYS LETT B, 1991
We construct the second hamiltonian structure of the KP hierarchy as a natural extension of the Gel'fand-Dickey brackets of the generalized KdV hierarchies. The first structure - which has been recently identified as W 1+∞-is coordinated with the second structure and arises as a trivial (generalized) cocycle. The second structure gives rise to a non linear algebra, denoted W KP, with generators of weights 1, 2, … . The reduced algebra obtained by setting the weight 1 field to zero contains a centerless Virasoro subalgebra, and we argue that this is a universal W algebra from which all W n algebras are obtained through reduction.
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
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.
1992
This paper is devoted to constructing a quantum version of the famous KP hierarchy, by deforming its second Hamiltonian structure, namely the nonlinear hatWinfty\hat{W}_{\infty}hatWinfty algebra. This is achieved by quantizing the conformal noncompact SL(2,R)k/U(1)SL(2,R)_{k}/U(1)SL(2,R)k/U(1) coset model, in which hatWinfty\hat{W}_{\infty}hatWinfty appears as a hidden current algebra. For the quantum hatWinfty\hat{W}_{\infty}hatWinfty algebra at level k=1k=1k=1, we have succeeded in constructing an infinite
Poisson Groups and Differential Galois Theory of Schroedinger Equation on the Circle
Communications in Mathematical Physics, 2008
We combine the projective geometry approach to Schroedinger equations on the circle and differential Galois theory with the theory of Poisson Lie groups to construct a natural Poisson structure on the space of wave functions (at the zero energy level). Applications to KdV-like nonlinear equations are discussed. The same approach is applied to 2 nd order difference operators on a one-dimensional lattice, yielding an extension of the lattice Poisson Virasoro algebra.