The algebra of observables in noncommutative deformation theory (Q2285225)

\(A\) denotes a finite dimensional associative algebra over a field \(k\) and \(M_1, \dots ,M_r\) the (isomorphism classes of) simple right \(A\)-modules. If \(\mathrm{End}_A (M_i ) = k\) for all \(i\), then the algebra homomorphism \(\rho : A \to \bigoplus_i \mathrm{End}_k (M_i )\) is surjective, where \(\rho \) is given by right multiplication of \(A\) on \(M_i\). This classical result was extended by \textit{O. A. Laudal} [Homology Homotopy Appl. 4, No. 2(2), 357--396 (2002; Zbl 1013.16018)] to what is called the Generalized Burnside Theorem: He replaced \(\rho \) by the versal morphism \(\eta\) arising from the noncommutative deformation functor \(\mathrm{Def}_{\mathrm M}\), where \(\mathrm{M}=\{ M_1, \dots M_r\}\) denotes a swarm of right \(A\)-modules. Then \(\mathrm{Def}_{\mathrm M}\) has a pro-representing hull \(H\), and \(\mathcal O (\mathrm{M}):= \mathrm{End}_H(M_H) \cong (H_{ij}\otimes_k\mathrm{Hom}_k(M_i,M_j))\) for a versal family \(M_H\). \(\mathcal{O} (\mathrm{M})\) is said to be the algebra of observables of the swarm \(\mathrm M\). The algebra homomorphism \(\rho : A \to \bigoplus_i\mathrm{End}_k (M_i )\) (as it was previously defined) factors into the algebra homomorphisms \(\eta : A \to\mathcal{O} (\mathrm{M})\) and \((H_{ij}\otimes_k\mathrm{Hom}_k(M_i,M_j)) \to \bigoplus_i\mathrm{End}_k(M_i)\). Assume again the above case, i.e. \(A\) finite dimensional and \(\mathrm M\) is the family of simple \(A\)-modules. From Laudal's work follows that \(\eta\) is an isomorphism if \(k\) is algebraically closed. The authors of the paper under review give the following generalization to an arbitrary field \(k\). Their first main result (Theorem 5) states: The homomorphism \(\eta : A \to \mathcal{O} (\mathrm{M})\) is injective. If furthermore \(\mathrm{End}_A (M_i ) = k\) for all \(i\), then \(\eta \) is an isomorphism. Especially if \(k\) is algebraically closed, the previously cited result of Laudal is recovered. In the final section on applications, an example is given where \(\eta\) is injective but not an isomorphism. The second main result is Theorem 7: ``Let \(A\) be a finitely generated \(k\)-algebra, let \(\mathrm{M} = \{ M_1,\dots M_r\}\) be a family of finite dimensional \(A\)-modules, and let \(B=\mathcal{O} (\mathrm{M})\). Then the versal morphism \(\eta^B :B\to\mathcal{O}^B (\mathrm{M})\) of \(\mathrm M\), considered as a family of right \(B\)-modules, is an isomorphism.'' Thus \((A,\mathrm{M}) \mapsto (B,\mathrm{M})\) is a closure operation.