Let (A,∆) be a Hopf-von Neumann algebra and R be the unitary fundamental operator on A defined by Takesaki in [28]: R(a⊗ b) = ∆(b)(a⊗ 1). Then R12R23 = R23R13R12 (see lemma 4.9 of [28]). This operator R plays a vital role in the theory of duality for von Neumann algebras (see [28] or [2]). If V is a vector space over an arbitrary field k, we shall study what we have called the Hopf equation: R1...

Green's theorem states that the Hall algebra of category representations a quiver over finite field is twisted bialgebra. Considering instead categories orthogonal or symplectic leads to class modules algebra, called modules, which are also comodules. A module theoretic analogue theorem, describing compatibility and comodule structures, not known. In this paper we prove analogues in degenerate ...

If H is a Hopf algebra with bijective antipode and α, β ∈ AutHopf (H), we introduce a category HYD (α, β), generalizing both Yetter-Drinfeld modules and anti-Yetter-Drinfeld modules. We construct a braided T-category YD(H) having all the categories HYD (α, β) as components, which ifH is finite dimensional coincides with the representations of a certain quasitriangular T-coalgebra DT (H) that we...

We prove a Maschke type Theorem for the category of Doi-Hopf modules. In fact, we give necessary and sufficient conditions for the functor forgetting the C-coaction to be separable. This leads to a generalized notion of integrals. Our results can be applied to obtain Maschke type Theorems for Yetter-Drinfel’d modules, Long dimodules and modules graded by G-sets. Existing Maschke type Theorems d...

We study the notions of relative comonad and comodule over a relative comonad. We use thesenotions to give categorical semantics for the coinductive type families of streams and of infinitetriangular matrices and their respective cosubstitution operations in intensional Martin-Löf typetheory. Our results are mechanized in the proof assistant Coq. 1998 ACM Subject Classification ...

We introduce group corings, and study functors between categories of comodules over group corings, and the relationship to graded modules over graded rings. Galois group corings are defined, and a Structure Theorem for the G-comodules over a Galois group coring is given. We study (graded) Morita contexts associated to a group coring. Our theory is applied to group corings associated to a comodu...

Let k be a commutative ring, A a k-algebra, C an A-coring that is projective as a left A-module, ∗C the dual ring of C and Λ a right C-comodule that is finitely generated as a left ∗C-module. We give necessary and sufficient conditions for projectivity and flatness of a module over the endomorphism ring EndC(Λ). If C contains a grouplike element, we can replace Λ with A.

We introduce the notion of vertex coalgebra, a generalization of vertex operator coalgebras. Next we investigate forms of cocommutativity, coassociativity, skew-symmetry, and an endomorphism, D∗, which hold on vertex coalgebras. The former two properties require grading. We then discuss comodule structure. We conclude by discussing instances where graded vertex coalgebras appear, particularly a...