Central extensions of current algebras

TL;DR

This paper computes the second cohomology of current Lie algebras, focusing on classical and modular semisimple Lie algebras, providing foundational results in the study of their extensions.

Contribution

It offers the first calculations of second cohomology for certain classes of current Lie algebras, including classical simple, Zassenhaus, and modular semisimple types.

Findings

Computed second cohomology groups for current Lie algebras.

Identified specific cocycles and extension classes.

Provided techniques potentially useful for future research.

Abstract

This is an old paper put here for archeological purposes. We compute the second cohomology of current Lie algebras of the form $L\otimes A$, where $L$ belongs to some class of Lie algebras which includes classical simple and Zassenhaus algebras, and of some modular semisimple Lie algebras. The results are largely superseded by subsequent papers, though, perhaps, some tricks and observations used here remain of minor interest.