Communications in Mathematics |

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 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 O-operators and Nijenhuis operators on left-symmetric Rinehart algebras.

Source: arXiv.org:2010.00335

Volume: Volume 32 (2024), Issue 2 (Special issue: CIMPA schools "Nonassociative Algebras and related topics, Brazil'2023" and "Current Trends in Algebra, Philippines'2024")

Published on: April 22, 2024

Accepted on: February 29, 2024

Submitted on: December 6, 2023

Keywords: Mathematics - Rings and Algebras,Mathematics - Representation Theory,Primary 17E05, 53D17, Secondary 14B12, 06B15

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