Double Extensions of Multiplicative Restricted Hom-Lie Algebras

In this paper, we study the double extension of a restricted quadratic Hom-Lie algebra $(V,[\cdot,\cdot]_{V},\alpha_{V},B_{V})$, 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.