On classification and deformations of Lie-Rinehart superalgebras (Q6588443)

A Lie-Rinehart superalgebra over a field \(\mathbb{K}\) is a pair \((A,L)\) with \(A\) being an associative, supercommutative \(\mathbb{K}\)-superalgebra and \(L\) being a Lie superalgebra over \(\mathbb{K}\) that admits a compatible \(A\)-module structure. Further, there is necessarily a compatible even morphism of Lie superalgebras \(\rho: L \to \operatorname{Der}(A)\) called the anchor map, where \(\operatorname{Der}(A)\) is the Lie superalgebra of super derivations of \(A\).\N\NThe first goal of the paper is to give a classification of all Lie-Rinehart superalgebras over the complex numbers when the dimension of \(A\) is at most 2 and the dimension of \(L\) is at most 4. For any such pair, one can make a trivial construction by taking a trivial \(A\)-action and a null anchor map, so the task is to identify those cases that admit non-trivial constructions. This is completed on an essentially case-by-case basis, using the known classifications of \(A\) and \(L\). In some cases, computer calculations are needed. The authors ultimately give explicit descriptions of the actions and anchor maps in all the non-trivial cases.\N\NThe authors then consider a deformation theory (for \(\mathbb{K}\) having characteristic zero) following the work of \textit{A. Mandal} and \textit{S. K. Mishra} [Commun. Alg 48, No. 4, 1653--1670 (2020; Zbl 1480.17003)] for hom-Lie-Rinehart algebras. The deformation theory involves deforming the bracket operation on \(L\) and the anchor map but not the multiplication of \(A\) nor its action on \(L\). This theory requires the introduction of the notion of \textit{super-multiderivations} of \(L\) and a deformation cohomology complex built from these derivation spaces. A formal deformation of a Lie-Rinehart superalgebra \((A,L)\) is defined using power series (over \(\mathbb{K}\) and \(\mathbb{L}\)). Analogous to other contexts, it is shown that the \textit{infinitesimal} portion of a formal deformation is a 2-cocycle in this deformation cohomology. It is also shown that, if the second deformation cohomology of \(L\) is zero, then any deformation is trivial. A Lie-Rinehart superalgebra admitting only the trivial deformation is referred to as \textit{rigid}, and the authors show that one of their small-dimensional examples is in fact rigid.