Principal comodule algebras can be thought of as objects representing principal bundles in non-commutative geometry. A crucial component of a principal comodule algebra is a strong connection map. For some applications it suffices to prove that such a map exists, but for others, such as computing the associated bundle projectors or Chern–Galois characters, an explicit formula for a strong conne...

By a theorem due to Kato and Ohtake, any (not necessarily strict) Morita context induces an equivalence between appropriate subcategories of the module categories of the two rings in the Morita context. These are in fact categories of firm modules for non-unital subrings. We apply this result to various Morita contexts associated to a comodule Σ of an A-coring C. This allows to extend (weak and...

This is the second part of the article [3]. In the first paper we developed a cyclic homology theory for B–module coalgebras with coefficients in stable B–module/comodules where B was just a bialgebra. The construction we gave for the cyclic homology theory for B–module coalgebras used mainly the coalgebra structure on B. In the first part of this paper, we present the dual picture. Namely, a c...

For coalgebras C over a field, we study when the categories CM of left C-comodules and MC of right C-comodules are symmetric categories, in the sense that there is a duality between the categories of finitely presented unitary left R-modules and finitely presented unitary left L-modules, where R and L are the functor rings associated to the finitely accessible categories CM and MC .

Let M be a vector space over a field k and R ∈ End k(M ⊗ M). This paper studies what shall be called the Long equation: that is, the system of nonlinear equations R12R13 = R13R12 and R12R23 = R23R12 in End k(M ⊗ M ⊗ M). Any symmetric solution of this system supplies us a solution of the integrability condition of the Knizhnik-Zamolodchikov equation: [R12, R13+R23] = 0 ([4] or [10]). We shall ap...

We construct topological quantum field theories (TQFTs) and commuting projector Hamiltonians for any 1+1d gapped phases with non-anomalous fusion category symmetries, i.e. finite symmetries that admit SPT phases. The construction is based on two-dimensional state sum TQFT whose input datum an $H$-simple left $H$-comodule algebra, where $H$ a dimensional semisimple Hopf algebra. show the actions...

We show that semisimple Hopf algebras having a self-dual faithful irreducible comodule of dimension 2 are always obtained as abelian extensions with quotient Z2. We prove that nontrivial Hopf algebras arising in this way can be regarded as deformations of binary polyhedral groups and describe its category of representations. We also prove a strengthening of a result of Nichols and Richmond on c...

Any finite-dimensional Hopf algebra H is Frobenius and the stable category of H-modules is triangulated monoidal. To H-comodule algebras we assign triangulated module-categories over the stable category of H-modules. These module-categories are generalizations of homotopy and derived categories of modules over a differential graded algebra. We expect that, for suitable H, our construction could...

In this paper, we mainly investigate a revised induction funtor to study Hom-Doi-Hopf modules and the coinvariant for Hom-Hopf comodule algebras.