Hom-L-R-smash products, Hom-diagonal crossed products and the Drinfeld double of a Hom-Hopf algebra

Abstract

We introduce the Hom-analogue of the L-R-smash product and use it to define the Homanalogue of the diagonal crossed product. When H is a finite dimensional Hom-Hopf algebra with bijective antipode and bijective structure map, we define the Drinfeld double of H; its algebra structure is a Hom-diagonal crossed product and it has all expected properties, namely it is quasitriangular and modules over it coincide with left-right Yetter-Drinfeld modules over H.

The Drinfel’d double versus the Heisenberg double for Hom-Hopf algebras

“…In this section, we will recall the definitions in [8] on the Hom-Hopf algebras, Hommodules and Hom-comodules.…”

“…for any a ∈ A and h ∈ H. By Proposition 2.6 in [8], in order to prove the multiplication is Hom-associative, we need only to verify that T is a Hom-twisting map between H and A. Indeed, first easy to see that…”

“…In [8], the authors have defined the Hom-associative algebra (H * , (α −1 ) * ), the linear dual of H, where the multiplication is given by…”

Crossed products of Hom-Hopf algebras

“…In this section, we will recall the definitions in [20] on Hom-Hopf algebras, Hom-modules and Homcomodules.…”

Entwined Hom-modules and frobenius properties

“…Recall from [17] that let (A, β A ) and (B, β B ) be two Hom-associative algebras and a linear map R : A ⊗ B → B ⊗ A with R(b ⊗ a) = a R ⊗ b R = a r ⊗ b r satisfying the conditions:…”