Extensions for Generalized Current Algebras

TL;DR

This paper investigates extension groups for modules over generalized current algebras, providing formulas and complete descriptions for specific cases, especially for the algebra rak{sl}_2.

Contribution

It introduces formulas for Ext groups between simple modules over generalized current algebras and fully characterizes Ext^2 for rak{sl}_2 modules formed by evaluation modules.

Findings

Formulas for Ext^1 and Ext^2 between simple modules

Complete description of Ext^2 for rak{sl}_2 modules with evaluation modules

Finite-dimensionality conditions for extension groups

Abstract

Given a complex semisimple Lie algebra ${\mathfrak g}$ and a commutative ${\mathbb C}$-algebra $A$, let ${\mathfrak g}[A] = {\mathfrak g} \otimes A$ be the corresponding generalized current algebra. In this paper we explore questions involving the computation and finite-dimensionality of extension groups for finite-dimensional ${\mathfrak g}[A]$-modules. Formulas for computing $\operatorname{Ext}^{1}$ and $\operatorname{Ext}^{2}$ between simple ${\mathfrak g}[A]$-modules are presented. As an application of these methods and of the use of the first cyclic homology, we completely describe $\operatorname{Ext}^{2}_{{\mathfrak g}[t]}(L_{1},L_{2})$ for ${\mathfrak g}=\mathfrak{sl}_{2}$ when $L_{1}$ and $L_{2}$ are simple ${\mathfrak g}[t]$-modules that are each given by the tensor product of two evaluation modules.