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| Main Author: | |
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| Format: | Preprint |
| Published: |
2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2012.01561 |
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| _version_ | 1866908663582556160 |
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| author | Saadaoui, Nejib |
| author_facet | Saadaoui, Nejib |
| contents | This paper develops a cohomology theory for Hom-Leibniz algebras using the
$β$-Nijenhuis--Richardson bracket and applies it to classify non-abelian
extensions. We introduce left, and right versions of the bracket,
each defining a graded Lie algebra structure on the space of $β$-cochains.
The main result establishes that equivalence classes of split extensions of a
Hom-Leibniz algebra $L$ by $V$ are in bijection with the second cohomology
space $H^2(L,V)$, generalizing classical results from Lie and Leibniz algebra
theory. We characterize extensions explicitly through 2-cocycles $(λ_l,
λ_r, θ)$ and provide complete classifications of low-dimensional cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_01561 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Non-abelian cohomology and extensions of Hom-algebras via the $\boldsymbolβ$-Nijenhuis--Richardson bracket Saadaoui, Nejib Rings and Algebras This paper develops a cohomology theory for Hom-Leibniz algebras using the $β$-Nijenhuis--Richardson bracket and applies it to classify non-abelian extensions. We introduce left, and right versions of the bracket, each defining a graded Lie algebra structure on the space of $β$-cochains. The main result establishes that equivalence classes of split extensions of a Hom-Leibniz algebra $L$ by $V$ are in bijection with the second cohomology space $H^2(L,V)$, generalizing classical results from Lie and Leibniz algebra theory. We characterize extensions explicitly through 2-cocycles $(λ_l, λ_r, θ)$ and provide complete classifications of low-dimensional cases. |
| title | Non-abelian cohomology and extensions of Hom-algebras via the $\boldsymbolβ$-Nijenhuis--Richardson bracket |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2012.01561 |