Extensions of associative algebras

Abstract

In this paper, we translate the problem of extending an associative algebra by another associative algebra into the language of codifferentials. The authors have been constructing moduli spaces of algebras and studying their structure by constructing their versal deformations. The codifferential language is very useful for this purpose. The goal of this paper is to express the classical results about extensions in a form which leads to a simpler construction of moduli spaces of low-dimensional algebras.

The moduli space of 4-dimensional nilpotent complex associative algebras

“…There is a classical theory of extensions, which was developed by many contributors going back as early as the 1930s. In [2], we gave a description of the theory of extensions of an algebra W by an algebra M . Consider the diagram 0 → M → V → W → 0 of associative K-algebras, so that V = M ⊕ W as a K-vector space, M is an ideal in the algebra V , and W = V /M is the quotient algebra.…”

Gröbner-Shirshov bases theory and extensions of Leibniz superalgebras

“…There are many results on extensions of algebras, in particular, on extensions of associative algebras, Lie (super)algebras, Leibniz (super)algebras, etc, see, for example, [4,5,20,21,24,29,35]. Mostly, they deal with some special cases for extensions.…”

Extensions of associative and Lie algebras via Gröbner-Shirshov bases method

“…There are many results on extensions of algebras, in particular on extensions of associative algebras, Lie algebras, Lie superalgebras, etc, see, for example, [2,3,22,27,28,32,34,35,36,38,43]. Mostly, they deal with some special cases for extensions.…”