Cohomology and deformations of left-symmetric Rinehart algebras. (English) Zbl 07900714
Summary: We introduce a notion of left-symmetric Rinehart algebras, which is a generalization of a left-symmetric algebras. The left multiplication gives rise to a representation of the corresponding sub-adjacent Lie-Rinehart algebra. We construct left-symmetric Rinehart algebra from \(\mathcal{O}\)-operators on Lie-Rinehart algebra. We extensively investigate representations of a left-symmetric Rinehart algebras. Moreover, we study deformations of left-symmetric Rinehart algebras, which is controlled by the second cohomology class in the deformation cohomology. We also give the relationships between \(\mathcal{O}\)-operators and Nijenhuis operators on left-symmetric Rinehart algebras.
MSC:
| 17D25 | Lie-admissible algebras |
| 53D17 | Poisson manifolds; Poisson groupoids and algebroids |
| 17B70 | Graded Lie (super)algebras |
| 17B56 | Cohomology of Lie (super)algebras |
| 14B12 | Local deformation theory, Artin approximation, etc. |
Keywords:
left-symmetric Rinehart algebra; representation; graded Lie algebra; Maurer-Cartan element; cohomology; deformation; Nijenhuis operatorReferences:
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