Crossed modules for Lie-Rinehart algebras.
Let \(k\) be a field and \(A\) be a commutative \(k\)-algebra. A {Lie-Rinehart algebra} over \(A\) is a Lie \(k\)-algebra \(\mathcal{L}\) which also has the structure of an \(A\)-module. Further \(\mathcal{L}\) must admit an appropriately compatible Lie \(k\)-algebra and an \(A\)-module homomorphism to the set of \(k\)-derivations on \(A\). Note that any Lie \(A\)-algebra can be considered as a Lie-Rinehart algebra over \(A\) in a natural way. The authors introduce the notion of a {crossed module} of Lie-Rinehart algebras over \(A\). A crossed module consists of a pair \((\mathcal{L},\mathcal{R})\) where \(\mathcal{L}\) is a Lie-Rinehart algebra over \(A\) and \(\mathcal{R}\) is a Lie \(A\)-algebra. Further, \(\mathcal{L}\) is assumed to act on \(\mathcal{R}\) (i.e., there exists a map \(\mathcal{L}\otimes\mathcal{R} \to \mathcal{R}\) satisfying certain properties) and there exists a Lie \(k\)-algebra homomorphism \(\mathcal{R} \to \mathcal{L}\) which is well-behaved with respect to the actions of \(A\) and \(\mathcal{L}\). This definition extends the definition of crossed modules for a Lie algebra introduced by \textit{C. Kassel} and \textit{J.-L. Loday} [Ann. Inst. Fourier 32, 119--142 (1982; Zbl 0485.17006)]. The main goal of this paper is to generalize the work of Kassel and Loday to show that these newly defined crossed modules can be classified by the third cohomology of Lie-Rinehart algebras.
scientific article; zbMATH DE number 2066828
scientific article