On graded matrix Hom-algebras

Abstract

Abstract. Consider an (associative) matrix algebra M I (R) graded by means of an abelian group G, and a graded automorphism φ on M I (R). By defining a new product by x ⋆ y := φ(x)φ(y) on M I (R), (M I (R), ⋆) becomes a hom-associative algebra graded by a twist of G. The structure of (M I (R), ⋆) is studied, by showing that M I (R) is of the formwith U an R-submodule of the 0-homogeneous component and any I j a well described graded ideal of M I (R), satisfying I j ⋆ I k = 0 if j = k. Under certain conditions, the graded simplicity of an arbitrary graded hom-associative algebra M is characterized and it is shown that M is the direct sum of the family of its simple graded ideals.

BiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras

“…Since ξ = 0 and λ i = 0 for all i = 1, 2, 3, 4, all the maps α, β, ψ, ω, α A , β A are bijective. According to Theorem 8.1, the map R : U q (sl 2 ) (α,β,ψ,ω) ⊗A 2|0 q,α,β → A 2|0 q,α,β ⊗U q (sl 2 ) (α,β,ψ,ω) defined by (8.1) leads to the smash product A 2|0 q,α,β #U q (sl 2 ) (α,β,ψ,ω) whose multiplication is defined by (a#h)(a #h ) = a * β −1 ω −1 (h (1) ) β −1 A (a ) #ψ −1 (h (2) ) • h ,…”

“…In the last years, many concepts and properties from classical algebraic theories have been extended to the framework of Hom-structures, see for instance [2,3,7,8,11,16,20,22,25,26,27,31,30].…”