Highest-Weight Theory for Truncated Current Lie Algebras

TL;DR

This paper develops a highest-weight theory for truncated current Lie algebras, providing a reducibility criterion for Verma modules across various Lie algebras including Kac-Moody, Heisenberg, and Virasoro.

Contribution

It introduces a highest-weight framework for truncated current Lie algebras with a reducibility criterion for Verma modules, extending to several important classes of Lie algebras.

Findings

Established a reducibility criterion for Verma modules.

Applied the theory to Kac-Moody, Heisenberg, and Virasoro algebras.

Analyzed the Shapovalov form to achieve results.

Abstract

Let g denote a Lie algebra over a field of characteristic zero, and let T(g) denote the tensor product of g with a ring of truncated polynomials. The Lie algebra T(g) is called a truncated current Lie algebra, or in the special case when g is finite-dimensional and semisimple, a generalized Takiff algebra. In this paper a highest-weight theory for T(g) is developed when the underlying Lie algebra g possesses a triangular decomposition. The principal result is the reducibility criterion for the Verma modules of T(g) for a wide class of Lie algebras g, including the symmetrizable Kac-Moody Lie algebras, the Heisenberg algebra, and the Virasoro algebra. This is achieved through a study of the Shapovalov form.