The algebra of q-pseudodifferential symbols and the q-W_{KP}^n algebra

Abstract

In this paper we continue with the program to explore the topography of the space of W-type 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.

28 pages. Section 5 revised. Misprints corrected. References added

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Cited by in corpus (12)

• The Extended Bigraded Toda hierarchy
• q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy
• A new extended q-deformed KP hierarchy
• The q-deformed mKP hierarchy with self-consistent sources, Wronskian solutions and solitons
• Quantum Torus symmetry of the KP, KdV and BKP hierarchies
• $q$-analogue of modified KP hierarchy and its quasi-classical limit
• Sato theory on the $q$-Toda hierarchy and its extension
• Integrable discrete systems on R and related dispersionless systems
• Local and non-local multiplicative Poisson vertex algebras and differential-difference equations
• Virasoro and W-constraints for the $q$-KP hierarchy
• The generalised q-Wronskian solutions of the q-deformed constrained modified KP hierarchy
• The Discrete AKNS-D Hierarchy