Double Extensions of Multiplicative Restricted Hom-Lie Algebras
Dan Mao, Zeyu Hao, Liangyun Chen
TL;DR
This work extends double extension theory to finite-dimensional restricted quadratic Hom-Lie algebras in prime characteristic. By constructing $L=\mathscr{E}^{*}\oplus V\oplus\mathscr{E}$ from a restricted quadratic Hom-Lie algebra $(V,[\cdot,\cdot]_{V},\alpha_V,B_V)$ with a $\mathscr{D}$-invariant form, the authors prove that the resulting structure is a multiplicative quadratic Hom-Lie algebra and that the associated $2$-structure (for $p=2$) or $p$-structure (for $p>2$) extends consistently to $L$; they establish necessary invertibility and involution criteria for the twist map $\alpha$. They provide converse results: any irreducible restricted quadratic Hom-Lie algebra with nonzero center arises as such a double extension, and they develop adapted isomorphisms and classifications of the resulting $p$-structures. In addition, the paper presents concrete examples in characteristic $2$ and discusses limitations due to lacking suitable cohomology theories, pointing to future work on restricted Hom-Lie cohomology to generate more examples and a deeper classification theory.
Abstract
In this paper, we study the double extension of a restricted quadratic Hom-Lie algebra $(V,[\cdot,\cdot]_{V},α_{V},B_{V})$, which is an enlargement of $V$ by means of a central extension and a restricted derivation $\mathscr{D}$. In particular, we prove that the double extension of a restricted quadratic Hom-Lie algebra $V$ with a $\mathscr{D}$-invariant bilinear form $B_{V}$ is restricted. Conversely, any irreducible restricted quadratic Hom-Lie algebra with nonzero center is proved to be the double extension of another restricted quadratic Hom-Lie algebra.