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.