Classical Extended Conformal Algebras Associated with Constrained KP Hierarchy

TL;DR

This paper explores the conformal properties of the second Hamiltonian structure of the constrained KP hierarchy, revealing a family of nonlocal extended conformal algebras and their relation to known algebraic structures.

Contribution

It introduces a new family of nonlocal extended conformal algebras associated with the constrained KP hierarchy and connects them to existing algebraic frameworks like W-U(1)-Kac-Moody.

Findings

Identified nonlocal extended conformal algebras from the second Hamiltonian structure.

Mapped the constrained KP hierarchy to Kuperschmidt's Lax hierarchy and analyzed the Hamiltonian structure.

Showed the Hamiltonian structure defines the W-U(1)-Kac-Moody algebra.

Abstract

We examine the conformal property of the second Hamiltonian structure of constrained KP hierarchy derived by Oevel and Strampp. We find that it naturallygives a family of nonlocal extended conformal algebras. We give two examples of such algebras and find that they are similar to Bilal's V algebra. By taking a gauge transformation one can map the constrained KP hierarchy to Kuperschmidt's nonstandard Lax hierarchy. We consider the second Hamiltonian structure in this representation. We show that after mapping the Lax operator to a pure differential operator the second structure becomes the sum of the second and the third Gelfand-Dickey brackets defined by this differential operator. We show that this Hamiltonian structure defines the W-U(1)-Kac-Moody algebra by working out its conformally covariant form.