The moduli space of 4-dimensional non-nilpotent complex associative algebras (Q2343350)

The classification of associative algebras started with Benjamin Peirce who gave a partial classification of the complex associative algebras of dimension up to 6. This classification relied on the fact that every finite dimensional algebra which is not nilpotent contains a nontrivial idempotent element. This result is a main ingredient in the fundamental theorem of finite dimensional associative algebras: Theorem. Let \(A\) be finite dimensional over \(\mathbb K\) with radical \(N\). Then \(A/N\) is semisimple, i.e. a direct sum of simple algebras. The result also leads to Wedderburn's theorem: Theorem. A finite dimensional algebra over \(\mathbb K\) is simple if and only if \(A\simeq M\otimes D\) where \(M=\mathfrak{gl}(n,\mathbb K)\) and \(D\) is a division algebra over \(\mathbb K\). Over \(\mathbb C\) the only division algebra is \(\mathbb C\) it self, so Wedderburn's theorem states that the only simple algebras are the matrix algebras. There is exactly one simple \(4\)-dimensional complex associative algebra \(\mathfrak{gl}(2,\mathbb C)\), and there is one additional semisimple algebra which is the direct sum of 4 copies of \(\mathbb C\). The authors mentions two prior approaches to the classification. The first, by \textit{B. Peirce} [Am. J. Math. 4, 99--230 (1881; JFM 13.0082.03)], includes mistakes, the second, by \textit{G. Mazzola} [Comment. Math. Helv. 55, 267--293 (1980; Zbl 0463.14004)], classifies only unital algebras, and that turns out to not be sufficient. Given an effective method of constructing all unital algebras of arbitrary dimension, and to determine their maximal nilpotent ideals, the authors are able to recover all nilpotent algebras of dimension \(n\) from their enlargements. To recover all algebras of dimension \(n\), they would only have to consider extensions of nilpotent algebras of dimension \(k\) by semisimple algebras of dimension \(n-k\). The method used in this article is effective for constructing extensions of nilpotent algebras by semisimple ones. The article's main issue is to explore the construction method which leads to the description of all algebras, and the main goal is to give a complete description of the moduli space of non-nilpotent \(4\)-dimensional associative algebras, including a computation of the miniversal deformation of every element. The description is obtained by extensions, and this is the new thing in this article. The discussion ends with a canonical stratification of the moduli space into projective orbifolds of a simple type such that the strata are connected only by deformations factoring through jump deformations, and the elements of a particular stratum are given by neighbourhoods determined by smooth deformations. The article is explicit and detailed. To illustrate this, the construction of algebras by extensions is given by the following: Given an exact sequence of associative \(\mathbb K\)-algebras \(0\rightarrow M\rightarrow V\rightarrow W\rightarrow 0\). Then \(V=M\oplus W\), \(M\) is an ideal in \(V\) and \(W=V/M\) is the quotient. If \(\delta\in C^2(W)\) and \(\mu\in C^2(M)\) represent the algebra structures on \(W\) and \(M\) respectively, \(\mu,\delta\) can be seen as elements in \(C^2(V)\). Let \(T^{k,l}\) be the subspace of \(T^{k+l}(V)\) given recursively by \(T^{0,0}=\mathbb K\), \(T^{k,l}=M\oplus T^{k-1,l}\oplus V\oplus T^{k,l-1}\). If \(d\) is the algebra structure on \(V\), \(d=\delta+\mu+\lambda+\psi\), \(\lambda\in C^{1,1}\), \(\psi\in C^{0,2}\). The condition of associativity on \(V\) gives several conditions following from \([d,d]=0\), and this finally induces that the general group of equivalences of extensions of the algebra structure \(\delta\) on \(W\) by the algebra structure \(\mu\) on \(M\) is given by the group of automorphisms of \(V\) of the form \(h=\exp(\beta)g\), \(\beta\in C^{0,1}\), \(g\in G_{\delta,\mu}\) and the authors use this to classify the extensions up to equivalence. Also, the fundamental theorem of finite dimensional algebras allows to restrict the consideration to two cases when looking at extensions of semisimple algebras by nilpotent algebras which is what is necessary to classify all non-nilpotent algebras. I would like to point out that the article contains a really good treatment on stratifications of moduli spaces (of algebras) and of jump deformations. Also, there is an interesting example of a commutative algebra deforming into a non-commutative one. The article is very well written, and serves as a reference on the 4-dimensional non-nilpotent complex associative algebras. It gives explicit tables of all cases.