Central extensions of current algebras

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 The results are largely superseded by subsequent papers, though, perhaps, some tricks and observations used here remain of minor interest.

💡 Research Summary

The paper investigates central extensions of current Lie algebras, i.e., Lie algebras of the form (L\otimes A) where (L) is a Lie algebra and (A) a commutative associative algebra. The main objective is to compute the second Lie‑algebra cohomology group (H^{2}(L\otimes A,\mathbb{K})), because central extensions are classified by this group. The author focuses on a class of Lie algebras (L) that includes the classical simple algebras (such as (\mathfrak{sl}{n},\mathfrak{so}{n},\mathfrak{sp}_{n})) and the Zassenhaus algebras, and also treats certain modular semisimple algebras in characteristic (p>0).

The methodology relies on the Hochschild‑Serre spectral sequence associated with the short exact sequence (0\to L\otimes \mathfrak{m}\to L\otimes A\to L\otimes (A/\mathfrak{m})\to0), where (\mathfrak{m}) is an ideal of (A). By analysing the (E_{2})‑page, the author obtains a decomposition \