Characterization of the $W_{1+\infty}$-n-algebra and applications

TL;DR

The paper develops a generalized framework for $W_{1+\infty}$ symmetry using the $\mathcal{R}(p,q)$-deformation, constructing $W_{1+\infty}$-$n$ algebras and their multi-variable extensions. It provides two complementary operator realizations, derives deformed commutators and $n$-brackets, and analyzes central extensions and Virasoro subalgebras, with attention to deformed Jacobi identities for even $n$. In matrix-model applications, it introduces a generalized theta function and generating function to obtain $W_{\infty}$ constraints for elliptic Hermitian models and a toy model, elucidating when $n$-brackets close and when abelian currents arise. Overall, the work reveals a broad class of deformed higher algebras and their constraint structures, offering tools for exploring integrable systems and generalized quantum algebras in multiple variables.

Abstract

In this paper, we construct the $W_{1+\infty}$-n-algebras in the framework of the generalized quantum algebra. We characterize the $\mathcal{R}(p,q)$-multi-variable $W_{1+\infty}$-algebra and derive its $n$-algebra which is the generalized Lie algebra for $n$ even. Furthermore, we investigate the $\mathcal{R}(p,q)$-elliptic hermitian matrix model and determine a toy model for the generalized quantum $W_{\infty}$ constraints. Also, we deduce particular cases of our results.

Theorems & Definitions (11)

• Remark 2.1
• Example 2.2
• Example 2.3
• Remark 2.4
• Remark 3.1
• Remark 3.2
• Remark 4.1
• Remark 5.1
• Remark 5.2
• Example 6.1