On Duality for a Braided Cross Product

In this note, we generalize a result of [4] (see also [9]) and set the isomorphism between the iterated cross product algebra H∨#(H#A) and braided analog of an A-valued matrix algebra H∨⊗A⊗H for a Hopf algebra H in the braided category C and for an H-module algebra A. As a preliminary step, we prove the equivalence between categories of modules over both algebras and category whose objects are Hopf Hmodules and A-modules satisfying certain compatibility conditions. Introduction and preliminaries A purpose of this note is to generalize the result of [4] (see also [9]) about the isomorphism between the iterated cross product algebra H∗#(H#A) and A-valued matrix algebra M(H)⊗A (for an H-module algebra A) to the fully braided case. Throughout this paper, the symbol C = (C,⊗, 1I) denotes a strict monoidal category with braiding Ψ. For convenience of the reader, we recall the necessary facts about braided monoidal categories and Hopf algebras in them. For object X ∈ C, we say that X∨ and ∨X ∈ C are dual objects if evaluation and coevaluation morphisms ev : X ⊗X∨ → 1I = ✒ ✑ X X ∨ , ev : ∨X ⊗X → 1I = ✒ ✑ ∨X X , coev : 1I → X∨ ⊗X = ✏ X∨ X , coev: 1I → X ⊗ ∨X = ✏ X ∨X can be chosen so that the compositions X = X ⊗ 1I 1⊗coev −−−−→ X ⊗ (X∨ ⊗X) = (X ⊗X∨)⊗X ev⊗1 −−−→ 1I⊗X = X, X = 1I⊗X coev⊗1 −−−−→ (X ⊗ ∨X)⊗X = X ⊗ (∨X ⊗X) 1⊗ev −−−→ X ⊗ 1I = X, X∨ = 1I⊗X∨ coev⊗1 −−−−→ (X∨ ⊗X)⊗X∨ = X∨ ⊗ (X ⊗X∨) 1⊗ev −−−→ X∨ ⊗ 1I = X∨, ∨X = ∨X ⊗ 1I 1⊗coev −−−−→ ∨X ⊗ (X ⊗ ∨X) = (∨X ⊗X)⊗ ∨X ev⊗1 −−−→ 1I⊗ ∨X = ∨X are all identity morphisms.