Abstract
In this paper, the universal enveloping algebra of color hom-Lie algebras is studied. A construction of the free involutive hom-associative color algebra on a hom-module is described and applied to ...
Highlights
The investigations of various quantum deformations of Lie algebras started a period of rapid expansion in 1980s, stimulated by the introduction of quantum groups motivated by applications to the quantum Yang–Baxter equation, quantum inverse scattering methods, and constructions of the quantum deformations of universal enveloping algebras of semisimple Lie algebras
It has been discovered in particular that in these q -deformations of Witt and Visaroro algebras and some related algebras, some interesting q deformations of Jacobi identities, extending Jacobi identity for Lie algebras, are satisfied. This has been one of the initial motivations for the development of general quasideformations and discretizations of Lie algebras of vector fields using more general σ -derivations in [25], and introduction of abstract quasi-Lie algebras and subclasses of quasi-hom-Lie algebras and hom-Lie algebras as well as their general colored counterparts in [25, 36, 37, 39, 58]. These generalized Lie algebra structures with twisted skew-symmetry and twisted Jacobi conditions by linear maps are tailored to encompass within the same algebraic framework such quasideformations and discretizations of Lie algebras of vector fields using σ derivations, describing general descritizations and deformations of derivations with twisted Leibniz rule, and the well-known generalizations of Lie algebras such as color Lie algebras, which are the natural generalizations of Lie algebras and Lie superalgebras
The Poincaré–Birkhoff–Witt theorem we prove a Poincaré–Birkhoff–Witt like type theorem for involutive color hom-Lie algebras
Summary
The investigations of various quantum deformations (or q -deformations) of Lie algebras started a period of rapid expansion in 1980s, stimulated by the introduction of quantum groups motivated by applications to the quantum Yang–Baxter equation, quantum inverse scattering methods, and constructions of the quantum deformations of universal enveloping algebras of semisimple Lie algebras. Making use of free involutive hom-associative algebras, the authors in [24] have found an explicit constructive way to obtain the universal enveloping algebras of hom-Lie algebras in order to prove the Poincaré–Birkhoff–Witt theorem. The following lemma shows an easy way to construct the universal algebra when we have an involutive color hom-Lie algebra. Let f : (g, [, ]g, βg) → (A, [, ]A, βA) be a morphism of color hom-Lie algebras and let B be the hom-associative subcolor algebra of A generated by f (g). (iii) In order to verify the universal property of (U (g), φg) in Definition 3.2, we only need to consider involutive hom-associative algebras A := (A, ·A, αA).
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