Comodule algebras of a Hopf algebroid H with a bijective antipode, i.e. algebra extensions B ⊆ A by H, are studied. Assuming that a lifted canonical map is a split epimorphism of modules of the (non-commutative) base algebra of H, relative injectivity of the H-comodule algebra A is related to the Galois property of the extension B ⊆ A and also to the equivalence of the category of relative Hopf...

If H is a quasi-Hopf algebra and B is a right H-comodule algebra such that there exists v : H → B a morphism of right H-comodule algebras, we prove that there exists a left Hmodule algebra A such that B ≃ A#H . The main difference comparing to the Hopf case is that, from the multiplication of B, which is associative, we have to obtain the multiplication of A, which in general is not; for this w...

We extend our work in [1] to the case of Hopf comodule coalgebras. We introduce the cocylindrical module C♮H, where H is a Hopf algebra with bijective antipode and C is a Hopf comodule coalgebra over H. We show that there exists an isomorphism between the cocyclic module of the crossed coproduct coalgebra C>◭H and ∆(C♮H), the cocyclic module related to the diagonal of C♮H. We approximate HC(C >...

To any cleft Hopf Galois object, i.e., any algebra H obtained from a Hopf algebra H by twisting its multiplication with a two-cocycle α, we attach two “universal algebras” AH and U α H . The algebra A α H is obtained by twisting the multiplication of H with the most general two-cocycle σ formally cohomologous to α. The cocycle σ takes values in the field of rational functions on H. By construct...

In this paper we examine a variety of algebraic contexts in which the quantum Yang–Baxter equation arises, and derive methods for generating new solutions from given ones. The solutions we describe are encoded in objects which have a module and a comodule structure over a bialgebra. Our work here is based in part on the ideas of [DR1, DR2]. Suppose that R : M ⊗M −→ M ⊗M is a solution to the qua...

Let X be a simplicial set. We construct a novel adjunction between the categories RX of retractive spaces over X and ComodX+ of X+comodules, then apply recent work on left-induced model category structures [5], [16] to establish the existence of a left proper, simplicial model category structure on ComodX+ with respect to which the adjunction is a Quillen equivalence after localization with res...

We introduce the notion of a partial corepresentation given Hopf algebra H over coalgebra C and closely related concept H-comodule. prove that there exists universal Hpar, associated to original H, such category regular H-comodules is isomorphic Hpar-comodules. coalgebroid show Hpar has structure suitable coalgebra.