Extensions for Generalized Current Algebras

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 ${\mathrm{Ext}}^1$ and ${\mathrm{Ext}}^2$ 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 ${\mathrm{Ext}}_{\mathfrak{g}[t]}^2(L_1,L_2)$ 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.

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