Subalgebras of $W_{1+\infty}$ and Their Quasifinite Representations

TL;DR

This paper introduces new polynomial-parametrized subalgebras of the $W_{1+ abla}$ algebra, explores their quasifinite representations, and examines their connections with $ ext{gl}( abla)$, including free field realizations and Kac determinants.

Contribution

It defines a family of subalgebras of $W_{1+ abla}$ parametrized by polynomials and analyzes their quasifinite representations and relations to $ ext{gl}( abla)$.

Findings

Defined new subalgebras parametrized by polynomials.

Analyzed quasifinite representations of these subalgebras.

Provided free field realizations and Kac determinants for the $ ext{Win}$ case.

Abstract

We propose a series of new subalgebras of the $W_{1+\infty}$ algebra parametrized by polynomials $p(w)$, and study their quasifinite representations. We also investigate the relation between such subalgebras and the $\hat{\mbox{gl}}(\infty)$ algebra. As an example, we investigate the $\Win$ algebra which corresponds to the case $p(w)=w$, presenting its free field realizations and Kac determinants at lower levels.