Quasifinite highest weight modules over the Lie algebra of differential operators on the circle

TL;DR

This paper classifies positive energy representations with finite degeneracies of the Lie algebra W_{1+∞} and constructs them using the representation theory of infinite matrix Lie algebras, also addressing unitary cases.

Contribution

It provides a classification and explicit construction of certain highest weight modules over the Lie algebra of differential operators on the circle, extending understanding of their structure.

Findings

Classified positive energy representations with finite degeneracies.

Constructed representations via infinite matrix Lie algebras.

Extended results to sin-algebras.

Abstract

We classify positive energy representations with finite degeneracies of the Lie algebra $W_{1+\infty}\/$ and construct them in terms of representation theory of the Lie algebra $\hatgl ( \infty R_m )\/$ of infinite matrices with finite number of non-zero diagonals over the algebra $R_m = \C [ t ] / ( t^{m + 1} )\/$. The unitary ones are classified as well. Similar results are obtained for the sin-algebras.