$W_n^{(\ka)}$ algebra associated with the Moyal KdV Hierarchy

TL;DR

This paper introduces a new algebraic structure, the $W_n^{(\u03ba)}$ algebra, linked to the Moyal KdV hierarchy, revealing its bi-Hamiltonian structure and free-field realization, with detailed examples for specific cases.

Contribution

It defines the $W_n^{(\u03ba)}$ algebra associated with the Moyal KdV hierarchy, extending the understanding of algebraic structures in integrable systems.

Findings

The $W_n^{(\u03ba)}$ algebra contains the Virasoro algebra with a specific central charge.

Explicit free-field realizations are constructed via Miura transformation.

Detailed cases for $W_3^{(\u03ba)}$ and $W_4^{(a)}$ are provided.

Abstract

We consider the Gelfand-Dickey (GD) structure defined by the Moyal $\star$-product with parameter $\ka$, which not only defines the bi-Hamiltonian structure for the generalized Moyal KdV hierarchy but also provides a $W_n^{(\ka)}$ algebra containing the Virasoro algebra as a subalgebra with central charge $\ka^2(n^3-n)/3$. The free-field realization of the $W_n^{(\ka)}$ algebra is given through the Miura transformation and the cases for $W_3^{(\ka)}$ and $W_4^{(\ka)}$ are worked out in detail.