Universal Enveloping Algebras of Hom-Lie Algebras with α-Derivations
Abstract
In this paper, we construct nonunital universal enveloping Hom-associative algebras for multiplicative Hom-Lie algebras equipped with α-derivations. We prove that the prescribed derivation extends to the enveloping algebra and that the resulting universal property yields a left adjoint to the commutator functor between the corresponding categories. When the twisting map is bijective, we establish a canonical isomorphism between this enveloping algebra and the twist of the nonunital universal enveloping algebra of the associated untwisted Lie algebra. This isomorphism preserves the twisting maps and the derivations. Consequently, the classical Poincaré-Birkhoff-Witt theorem identifies the associated graded algebra for the untwisted product with the positive-degree symmetric algebra. When the Hom-Lie algebra is finite-dimensional, the resulting PBW basis consists of ordered monomials of positive length. The twisting map, the extended α-derivation, and its untwisted counterpart preserve the PBW filtration.
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BibTeX
@misc{jiang2026universal,
title = {{Universal Enveloping Algebras of Hom-Lie Algebras with $α$-Derivations}},
author = {Zhangqiuyu Jiang and Chuangchuang Kang and Jiafeng Lü},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23522},
eprint = {2609.23522},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23522},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Universal Enveloping Algebras of Hom-Lie Algebras with α-Derivations
AU - Zhangqiuyu Jiang
AU - Chuangchuang Kang
AU - Jiafeng Lü
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23522
UR - https://arxiv.org/abs/2609.23522
AB - In this paper, we construct nonunital universal enveloping Hom-associative algebras for multiplicative Hom-Lie algebras equipped with α-derivations. We prove that the prescribed derivation extends to the enveloping algebra and that the resulting universal property yields a left adjoint to the commutator functor between the corresponding categories. When the twisting map is bijective, we establish a canonical isomorphism between this enveloping algebra and the twist of the nonunital universal enveloping algebra of the associated untwisted Lie algebra. This isomorphism preserves the twisting maps and the derivations. Consequently, the classical Poincaré-Birkhoff-Witt theorem identifies the associated graded algebra for the untwisted product with the positive-degree symmetric algebra. When the Hom-Lie algebra is finite-dimensional, the resulting PBW basis consists of ordered monomials of positive length. The twisting map, the extended α-derivation, and its untwisted counterpart preserve the PBW filtration.
ER -