Projections of Singular Vectors of Verma Modules over Rank 2 Kac-Moody Lie Algebras

TL;DR

This paper derives an explicit formula for projecting singular vectors in Verma modules over rank 2 Kac-Moody Lie algebras onto certain subalgebras, extending previous work with more explicit results.

Contribution

It provides a new explicit projection formula for singular vectors in Verma modules over rank 2 Kac-Moody Lie algebras, generalizing earlier less explicit formulas.

Findings

Derived an explicit projection formula for singular vectors

Extended previous results to more general rank 2 cases

Connected the formula to earlier work by Feigin, Fuchs, and Malikov

Abstract

We prove an explicit formula for a projection of singular vectors in the Verma module over a rank 2 Kac-Moody Lie algebra onto the universal enveloping algebra of the Heisenberg Lie algebra and of $sl_{2}$ (Theorem 3). The formula is derived from a more general but less explicit formula due to Feigin, Fuchs and Malikov [Funct. Anal. Appl. 20 (1986), no. 2, 103-113]. In the simpler case of $A_{1}^{1}$ the formula was obtained in [Fuchs D., Funct. Anal. Appl. 23 (1989), no. 2, 154-156].