$\Hom$-Lie structures on Lie algebras

Abstract

We give a systematic description of $\Hom$-Lie structures on several classes of Lie algebras and of their relations with transposed Poisson structures. Our starting point is the observation that the left-hand side of the defining identity is an alternating trilinear map; this makes $\mathcal{H}(\LL)$, the set of $\Hom$-Lie structures of $\LL$, a subspace of $\End(\LL)$ and reduces its computation to increasing triples of basis vectors. From this we derive, in every finite dimension $n$, the lower bound $\dim \mathcal{H}(\LL)\geq n \dim \mathcal{C}(\LL)$ and the codimension bound $\binom{n}{3}\dim [\LL,[\LL,\LL]]$, together with a refinement of the latter, and we determine $\mathcal{H}(\LL)$ for nilpotent, oscillator and solvable algebras, for $\sltwo \oplus A_{n-3}$, for the Witt, Virasoro and Heisenberg--Virasoro algebras, for all three-dimensional algebras, for $M_n(\bF)$ and $T_n(\bF)$, for the Virasoro-like algebra and its $q$-analogue, and for the planar Galilean conformal algebra. By a result of Filippov, every $\frac12$-derivation, and more generally every $δ$-derivation with $δ\neq0,1$, of a Lie algebra is a $\Hom$-Lie structure; we observe that, more generally, for $δ\neq 0,1$ the $δ$-derivations of an algebra of any variety defined by identities of degree at most three (associative, Novikov, left-symmetric, Leibniz, \ldots) are $\Hom$-structures of the corresponding $\Hom$-variety. We show that the inclusion of the $\frac12$-derivations into the $\Hom$-Lie structures is far from being an equality by computing the graded $\Hom$-Lie structures of Witt-, Virasoro- and Schrödinger-type algebras and of their central extensions, for which we give an explicit lifting obstruction. Along the way we correct two multiplication tables from the literature that violate the Jacobi identity.