Integrable representations of root-graded Lie algebras

TL;DR

This paper investigates the structure of integrable representations of root-graded Lie algebras, connecting their module categories to associative algebras, and unifies previous research on map algebras and isotropic Lie algebras.

Contribution

It characterizes the category of integrable modules of root-graded Lie algebras and relates it to associative algebra modules, extending and unifying prior work.

Findings

Linked representation categories to associative algebras for specific Lie algebras

Determined algebra structures for map algebras and sl_n(A)

Unified previous theories on map and isotropic Lie algebras

Abstract

The essential feature of a root-graded Lie algebra L is the existence of a split semisimple subalgebra g with respect to which L is an integrable module with weights in a possibly non-reduced root system S of the same rank as the root system R of g. Examples include map algebras (maps from an affine scheme to g, S = R), matrix algebras like sl_n(A) for a unital associative algebra A (S = R = A_{n-1}), finite-dimensional isotropic central-simple Lie algebras (S properly contains R in general), and some equivariant map algebras. In this paper we study the category of representations of a root-graded Lie algebra L which are integrable as representations of g and whose weights are bounded by some dominant weight of g. We link this category to the module category of an associative algebra, whose structure we determine for map algebras and sl_n(A). Our results unify previous work of Chari and her collaborators on map algebras and of Seligman on isotropic Lie algebras.