Quasifinite Representations of Classical Lie subalgebras of $W_{\infty,p}$

TL;DR

This paper classifies irreducible quasifinite highest weight representations of certain subalgebras of the Lie algebra of differential operators on the circle, revealing their structure in terms of infinite matrices and classical Lie subalgebras.

Contribution

It provides a complete classification of quasifinite highest weight modules for specific subalgebras of $W_{ infty,p}$, including explicit realizations in terms of matrix Lie algebras.

Findings

Exactly two anti-involutions of the differential operator algebra are identified.

Irreducible quasifinite highest weight modules are classified for the fixed subalgebras.

Realizations of modules are given in terms of infinite matrices over polynomial quotient algebras.

Abstract

We show that there are exactly two anti-involution $\sigma_{\pm}$ of the algebra of differential operators on the circle that are a multiple of $p(t\partial_t)$ preserving the principal gradation ($p\in\CC[x]$ non-constant). We classify the irreducible quasifinite highest weight representations of the central extension $\hat{\D}_p^{\pm}$ of the Lie subalgebra fixed by $-\sigma_{\pm}$. The most important cases are the subalgebras $\hat{\D}_x^{\pm}$ of $W_{\infty}$, that are obtained when $p(x)=x$. In these cases we realize the irreducible quasifinite highest weight modules in terms of highest weight representation of the central extension of the Lie algebra of infinite matrices with finitely many non-zero diagonals over the algebra $\CC[u]/(u^{m+1})$ and its classical Lie subalgebras of $C$ and $D$ types.