发表机构
School of Mathematical Sciences Zhejiang Normal University（浙江师范大学数学与科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为带 $\alpha$-导子的乘法 Hom-Lie 代数构造非酉泛包络 Hom-结合代数，证明导子延拓及左伴随性，并在双射扭转下建立与未扭转情形的同构，进而由 PBW 定理给出有限维情形的基与滤过保持。
AI 中文摘要
本文为配备 $\alpha$-导子的乘法 Hom-Lie 代数构造了非酉泛包络 Hom-结合代数。我们证明了给定的导子可延拓至包络代数，并且由此得到的泛性质给出了相应范畴之间换位子函子的左伴随。当扭转映射是双射时，我们建立了该包络代数与相关未扭转 Lie 代数的非酉泛包络代数的扭转之间的典范同构。该同构保持扭转映射和导子。因此，经典的 Poincaré-Birkhoff-Witt 定理将未扭转乘积的伴随分次代数等同于正次数对称代数。当 Hom-Lie 代数是有限维时，所得的 PBW 基由正长度的有序单项式组成。扭转映射、延拓的 $\alpha$-导子及其未扭转对应物保持 PBW 滤过。
英文摘要
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.
