Year: 2026
Type: preprint
Abstract: Burde proved that the Lie algebra $\mathfrak{sl}_2(\mathbb{K})$, over a field $\mathbb{K}$ of characteristic $p > 2$, admits compatible left-symmetric algebra structures if and only if $p = 3$. Since the even part of the Lie superalgebra $\mathfrak{osp}(1|2)$ is isomorphic to $\mathfrak{sl}_2(\mathbb{K})$, we ask whether Burde's structures can be extended to compatible left-symmetric superalgebra structures on $\mathfrak{osp}(1|2)$. We verify Burde's left-symmetric structures on $\mathfrak{sl}_2(\mathbb{K})$, and then test their possible extensions to $\mathfrak{osp}(1|2)$. We show that no such extension exists in characteristic 3. Together with the obstruction coming from the even part for characteristic $0$ and $p > 3$, this gives non-existence for a field $\mathbb{K}$ of characteristic different from $2$.
Source: arXiv (Cornell University)
Author: Loza, Josemaria
Language: English
Topic: Advanced Topics in Algebra
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221503204