Abstract

We study deformations and the moduli space of 3-dimensional complex associative algebras. We use extensions to compute the moduli space, and then give a decomposition of this moduli space into strata consisting of complex projective orbifolds, glued together through jump deformations. The main purpose of this paper is to give a logically organized description of the moduli space, and to give an explicit description of how the moduli space is constructed by extensions.

Highlights

  • The classification of associative algebras was instituted by Benjamin Peirce in the 1870’s [11], who gave a partial classification of the complex associative algebras of dimension up to 6, in some sense, one can deduce the complete classification from his results, with some additional work

  • In the examples of complex moduli spaces of Lie and associative algebras which we have studied, it turns out that there is a natural stratification of the moduli space of n-dimensional algebras by orbifolds, where the codifferentials on a given strata are connected by smooth deformations, which do not factor through jump deformations

  • One might say that in this paper, we prove this conjecture is true for the moduli space of 3-dimensional complex associative algebras

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Summary

Introduction

The classification of associative algebras was instituted by Benjamin Peirce in the 1870’s [11], who gave a partial classification of the complex associative algebras of dimension up to 6, in some sense, one can deduce the complete classification from his results, with some additional work. Every finite dimensional algebra which is not nilpotent contains a nontrivial idempotent element. A nilpotent algebra A is one which satisfies An = 0 for some n, while an idempotent element a satisfies a2 = a This observation of Peirce eventually leads to two important theorems in the classification of finite dimensional associative algebras. If A is not nilpotent, A/N is a semisimple algebra, that is, a direct sum of simple algebras. In the literature, the definition of a semisimple algebra is often given as one whose radical is trivial, and it is a theorem that semisimple algebras are direct sums of simple algebras. The main goal of this paper is to give a miniversal deformation of every element and give a complete description of the moduli space of 3-dimensional associative algebras. We will give a canonical stratification of the moduli space into projective orbifolds of a very simple type, so that the strata are connected only by deformations factoring through jump deformations, and the elements of a particular stratum are given by neighborhoods determined by smooth deformations

Preliminaries
Construction of algebras by extensions
Associative algebra structures on a 3-dimensional vector space
Extensions by the nontrivial nilpotent algebra
Extensions by the trivial nilpotent algebra
Deformations of the elements in the moduli space
Unital algebras
10 Commutative algebras
11 Conclusions
Full Text
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