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 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 ℋ(LL)≥ n dim 𝒞(LL) and the codimension bound binomn3dim [LL,[LL,LL]], together with a refinement of the latter, and we determine ℋ(LL) for nilpotent, oscillator and solvable algebras, for sltwo ⊕ An−3, for the Witt, Virasoro and Heisenberg--Virasoro algebras, for all three-dimensional algebras, for Mn(bF) and Tn(bF), for the Virasoro-like algebra and its q-analogue, and for the planar Galilean conformal algebra. By a result of Filippov, every 1/2-derivation, and more generally every δ-derivation with δ≠0,1, of a Lie algebra is a Hom-Lie structure; we observe that, more generally, for δ≠ 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 1/2-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.
Related
- RELATIVE BIHOM-PRE-ANTI-FLEXIBLE ALGEBRAS : CONSTRUCTIONS AND RELATED ALGEBRAIC STRUCTURES
- Extending structures for the linear framework of Universal Algebra with applications to transposed Poisson algebras
- Affinization of algebraic structures: Poisson algebras
- Novikov algebras and multi-indices in regularity structures
- mixed Hodge structures on the strictly upper triangular Lie algebras
BibTeX
@misc{adashev2026homlie,
title = {{$\Hom$-Lie structures on Lie algebras}},
author = {Jobir Adashev and Majidkhon Azizov and Vasily Voronin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32174},
eprint = {2609.32174},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32174},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hom-Lie structures on Lie algebras
AU - Jobir Adashev
AU - Majidkhon Azizov
AU - Vasily Voronin
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32174
UR - https://arxiv.org/abs/2609.32174
AB - 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 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 ℋ(LL)≥ n dim 𝒞(LL) and the codimension bound binomn3dim [LL,[LL,LL]], together with a refinement of the latter, and we determine ℋ(LL) for nilpotent, oscillator and solvable algebras, for sltwo ⊕ 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 1/2-derivation, and more generally every δ-derivation with δ≠0,1, of a Lie algebra is a Hom-Lie structure; we observe that, more generally, for δ≠ 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 1/2-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.
ER -