Abstract:Derived brackets provide a mechanism for generating algebraic structures from graded Lie superalgebras, with applications in Poisson geometry, mathematical physics, and the theory of algebroids. In this paper, we present a complete structural and isomorphism classification of a family of Lie algebras constructed from the general linear Lie superalgebra $\mathfrak{gl}_{m|n}$ over a field $\mathbb{K}$ of characteristic zero via the derived bracket generated by an odd element $B$ satisfying $B^2 = 0$, which endows $\mathfrak{g}_{-1}$ with a Lie algebra structure denoted $\mathfrak{g}_{-1}^{B}$. We prove that for fixed dimensions $m$ and $n$, the isomorphism type of $\mathfrak{g}_{-1}^{B}$ is entirely determined by $r=\operatorname{rank}(B)$. In arbitrary dimensions, two such algebras are isomorphic if and only if they share the same rank $r$ and satisfy $\{m,n\}=\{p,q\}$. We explicitly compute the Levi-Malcev decomposition, proving the semisimple Levi factor is isomorphic to $\mathfrak{sl}(r)$, and provide exact formulas for the solvable radical and center.
| Comments: | 21 pages. Poster presented at the 32st Colóquio Brasileiro de Matemática (IMPA, 2019) |
| Subjects: | Rings and Algebras (math.RA) |
| MSC classes: | 17B05, 17B70, 17B20, 17B30 |
| Cite as: | arXiv:2605.25470 [math.RA] |
| (or arXiv:2605.25470v1 [math.RA] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25470 arXiv-issued DOI via DataCite (pending registration) |
From: Luan Figueiredo De Oliveira [view email]
[v1]
Mon, 25 May 2026 06:20:05 UTC (14 KB)
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