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

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.

10 pages, LaTeX, RIMS-985, YITP/K-1076, YITP/U-94-22, SULDP-1994-4