arXiv subjects

A. Ben Hassine

Publications and source records attributed to A. Ben Hassine.

3 recordsLinked to original sources

arXiv2024

Cohomology and Deformations of left-symmetric Rinehart Algebras

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.

math.RA↗

arXiv2022

On Ternary $F$-manifold Algebras and their Representations

We introduce a notion of ternary $F$-manifold algebras which is a generalization of $F$-manifold algebras. We study representation theory of ternary $F$-manifold algebras. In particular, we introduce a notion of dual representation which requires additional conditions similar to the binary case. We then establish a notion of a coherence ternary $F$-manifold algebra. Moreover, we investigate the construction of ternary $F$-manifold algebras using $F$-manifold algebras. Furthermore, we introduce and investigate a notion of a relative Rota-Baxter operator with respect to a representation and use it to construct ternary pre-$F$-manifold algebras.

math.RA↗

arXiv2021

Extensions and crossed modules of $n$-Lie Rinehart algebras

We introduce a notion of $n$-Lie Rinehart algebras as a generalization of Lie Rinehart algebras to $n$-ary case. This notion is also an algebraic analogue of $n$-Lie algebroids. We develop representation theory and describe a cohomology complex of $n$-Lie Rinehart algebras. Furthermore, we investigate extension theory of $n$-Lie Rinehart algebras by means of $2$-cocycles. Finally, we introduce crossed modules of $n$-Lie Rinehart algebras to gain a better understanding of their third dimensional cohomology groups.

math.RA↗