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

We propose a series of new subalgebras of the W1+∞ algebra parametrized by polynomials p(w), and study their quasifinite representations. We also investigate the relation between such subalgebras and the ĝl(∞) algebra. As an example, we investigate the W∞ algebra which corresponds to the case p(w) = w, presenting its free field realizations and Kac determinants at lower levels.