Free involutive Hom-semigroups and Hom-associative algebras

Abstract

Abstract. In this paper we construct free Hom-semigroups when its unary operation is multiplicative and is an involution. Our method of construction is by bracketed words. As a consequence, we obtain free Hom-associative algebras generated by a set under the same conditions for the unary operation.

HNN-extension of involutive multiplicative Hom-Lie algebras

“…It is worth noting that there exists another approach provided in [80] for obtaining the universal enveloping algebra of a Hom-Lie algebra as a suitable quotient of the free Hom-nonassociative algebra through weighted trees, but the point of difficulty in the approach in [80] is the size of the weighted trees. Involutive Hom-Lie algebras have been constructed in [85], and the classical theory of enveloping algebras of Lie algebras was extended to an explicit construction of the free involutive Hom-associative algebra on a Hom-module in order to obtain the universal enveloping algebra [42]. This construction leads to a Poincare-Birkhoff-Witt theorem for the enveloping associative algebra of an involutive Hom-Lie algebra.…”

Matching Hom-Setting of Rota-Baxter Algebras, Dendriform Algebras, and Pre-Lie Algebras

“…Recently, there have been several interesting developments of Hom-Lie algebras in mathematics and mathematical physics, including Hom-Lie bialgebras [2,3], quadratic Hom-Lie algebras [4], involutive Hom-semigroups [5], deformed vector fields and differential calculus [6], representations [7,8], cohomology and homology theory [9,10], Yetter-Drinfeld categories [11], Hom-Yang-Baxter equations [12][13][14][15][16], Hom-Lie 2-algebras [17,18], ðm, nÞ-Hom-Lie alge-bras [19], Hom-left-symmetric algebras [20], and enveloping algebras [21]. In particular, the Hom-Lie algebra on semisimple Lie algebras was studied in [22], and the Hom-Lie structure on affine Kac-Moody was constructed in [23].…”

Universal enveloping algebras and Poincaré-Birkhoff-Witt theorem for involutive Hom-Lie algebras

“…A recent paper [19] gave an explicit construction of free involutive Hom-associative algebras generated by a set, without having to go into the Hom-nonassociative realm. This construction makes it possible to adapt the classical theory of enveloping algebras of Lie algebras to the Hom setting, namely to construct the universal enveloping algebra of an involutive Hom-Lie algebra from free involutive Hom-associative algebra and to attempt for a Poincaré-Birkhoff-Witt type theorem for this enveloping algebra.…”

“…We carry out this approach in this paper. First in Section 2 we modify the construction in [19] to obtain the free involutive Hom-associative algebra on a Hom-module. Based on this we provide in Section 3 another construction of the enveloping algebra of involutive Hom-Lie algebras in addition to the one given in [18].…”