On the Associative Algebra Kernels and Obstruction

TL;DR

This paper explores the theory of abstract kernels across various algebraic structures, reformulates Hochschild's associative algebra kernel theorem for Lie algebras, and demonstrates the independence of 3-cocycles in associative algebras.

Contribution

It recasts Hochschild's kernel construction for Lie algebras and proves the independence of 3-cocycles in associative algebras, extending kernel theory.

Findings

Reformulation of Hochschild's kernel theorem for Lie algebras

Proof of 3-cocycle independence in associative algebras

Use of universal enveloping algebra to simplify derivation algebra constructions

Abstract

The theory of abstract kernels in non-trivial extensions for many kinds of algebraical objects, such as groups, rings and graded rings, associative algebras, Lie algebras, restricted Lie algebras, DG-algebras and DG-Lie algebras, has been widely studied since 1940's. Gerhard Hochschild firstly treats associative algebra as an generic type in the series of kernel problems. He proves the theorem of constructing kernel by presenting many tedious relations that may lost the readers today. In this paper, we shall illustrate the formulation and recast it for Lie algebra(-oid) kernels. We also prove the independence of 3-cocycle in the case of associative algebra. Finally, we use the universal enveloping algebra of Lie algebra to reduce the difficulty of a direct construction for the derivation algebras.