Abstract
In this paper, we introduce the Hom-algebra setting of the notions of matching Rota-Baxter algebras, matching (tri)dendriform algebras, and matching pre-Lie algebras. Moreover, we study the properties and relationships between categories of these matching Hom-algebraic structures.
Highlights
The origin of Hom-structures may be found in the study of Hom-Lie algebras which were first introduced by Hartwig, Larsson, and Silvestrov [1]
A Hom-Lie algebra is a triple ðL,1⁄2−,−, αÞ consisting of a kmodule L, a bilinear skew-symmetric bracket 1⁄2−,−: L ⊗ L ⟶ L and an algebra endomorphism α : L ⟶ L satisfying the following Hom-Jacobi identity: 1⁄2αðxÞ, 1⁄2y, z + 1⁄2αðyÞ, 1⁄2z, x + 1⁄2αðxÞ, 1⁄2x, y
Generalizing the well-known result that an associative algebra has a Lie algebra structure via the commutator bracket, we show that a compatible Hom-associative algebra has a compatible Hom-Lie algebra structure
Summary
It is shown that the matching Rota-Baxter algebra was motivated by the studies of associative Yang-Baxter equations, Volterra integral equations, and linear structure of Rota-Baxter operators [38]. The main purpose of this paper is to extend these matching algebraic structures to the Hom-algebra setting and study the connections between these categories of Hom-algebras These results give rise to the following commutative diagram of categories, see Figure 3. A matching Hom-Lie algebra is a k-module g equipped with a collection of binary operations 1⁄2,ω : g ⊗ g ⟶ g, ω ∈ Ω and a linear map p : g ⟶ g such that. A compatible Hom-Lie algebra is a k-module g together with a set of binary operations 1⁄2,ω : g ⊗ g ⟶ g, ω ∈ Ω and a linear map p : g ⟶ g such that 1⁄2x, xω = 0.
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