Cohomology and deformations of left-symmetric Rinehart algebras (Q6591997)

This paper under review introduces a notion of left-symmetric Rinehart algebras which is a generalization of the notion of left-symmetric algebras. A left-symmetric Rinehart algebra is a quadruple \((L, A, \cdot, \ell)\), where \((L, \cdot)\) is a left-symmetric algebra, \(A\) is a commutative associative algebra and \(\ell: L\rightarrow \text{Der}(A)\) is a linear map with some compatibility conditions. The relationship between left-symmetric Rinehart algebras and Lie Rinehart algebras is similar to that between left-symmetric algebras and Lie algebras. Given a left-symmetric Rinehart algebra \((L, A, \cdot, \ell)\). One can obtain a Lie Rinehart algebra \((L, A, [\cdot,\cdot], \ell)\), where \((L, [\cdot,\cdot])\) is the sub-adjacent Lie algebra by taking the commutator of \((A, \cdot)\). Moreover, left-symmetric Rinehart algebras can be constructed by \(\mathcal{O}\)-operators on Lie Rinehart algebras.\N\NThis paper also introduces the definition of representations of left-symmetric Rinehart algebras and gives the dual representation of a representation. A graded Lie algebra on the space of multi-derivations is constructed whose Maurer-Cartan elements are left-symmetric Rinehart algebras which give rise to a coboundary operator of the deformation complex of left-symmetric Rinehart algebras. Moreover, this paper investigates deformations of left-symmetric Rinehart algebras in detail, for example, formal deformations of a left-symmetric Rinehart algebra are controlled by the second cohomology class in the deformation cohomology, obstructions to the extension of deformations are controlled by the third cohomology class in the deformation cohomology, trivial deformations of left-symmetric Rinehart algebras are related with Nijenhuis operators on left-symmetric Rinehart algebras and so on. Finally, this paper also studies some connections between \(\mathcal{O}\)-operators and Nijenhuis operators on left-symmetric Rinehart algebras.