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Deformations of complex 3-dimensional associative algebras. (English) Zbl 1223.14012

The authors study the moduli space of 3-dimensional complex associative algebras using deformation theory. The classification of such algebras was initiated in the 19th century, using the fundamental fact that every finite dimensional algebra which is not nilpotent contains a nontrivial idempotent element.
To compute the moduli space of 3-dimensional associative algebras, the authors use extensions of algebras based on the “fundamental theorem of finite dimensional associative algebras”: Suppose that \(A\) is a finite dimensional algebra over a field \(\mathbb K\). Then \(A\) has a maximal nilpotent ideal \(N\), called its radical. If \(A\) is not nilpotent, then \(A/N\) is a semisimple algebra.
Then the “Wedderburn theorem” is needed to construct extensions: If \(A\) is a finite dimensional algebra over \(\mathbb K\), then \(A\) is simple if and only if \(A\) is isomorphic to a tensor product \(M\otimes D\), where \(M=\mathfrak{gl}(n,\mathbb K)\) and \(D\) is a division algebra over \(\mathbb K\).
The main goal of the article is to give a miniversal deformation of every element and give a complete description of the moduli space of \(3\)-dimensional associative algebras. The authors give (actually compute) a canonical stratification of the moduli space into projective orbifolds 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.
The authors give the construction of algebras by extensions explicitly, including among other properties the Maurer–Cartan equations. This implies that the coalgebra structure is used. Explicit codifferentials for associative algebra structures on a 3-dimensional vector space are given, and the different extensions of algebras of dimension \(d<3\) by the different algebras of dimension \(3-d\) of interest is given. A complete classification of the \(3\)-dimensional complex associative algebras by means of extensions is given in form of a table.
Then the deformation theory using Hochschild cohomology groups are used to compute the versal families for each element in the table, giving the authors the possibility to investigate the following conjecture by Fialowski and Penkava: The moduli space of Lie or associative algebras of a fixed finite dimension \(n\) is stratified by projective orbifolds, with jump deformations and smooth deformations factoring through jump deformations providing the only deformations between the strata. This article proves that this is true for the moduli space of 3-dimensional complex associative algebras, by direct computation.
The article is self contained and explicit and gives yet another proof of the strength of deformation theory.

MSC:

14D15 Formal methods and deformations in algebraic geometry
13D10 Deformations and infinitesimal methods in commutative ring theory
14B12 Local deformation theory, Artin approximation, etc.
16S80 Deformations of associative rings
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
17B55 Homological methods in Lie (super)algebras
17B70 Graded Lie (super)algebras
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