Related papers: Duality Theorem and Drinfeld Double in Braided Ten…

In general, universal (co)measuring (co)monoids and universal (co)acting bi/Hopf monoids, which prove to be a useful tool in the classification of quantum symmetries, do not always exist. In order to ensure their existence, the support of a…

Let $H_1$ and $H_2$ be Hopf algebras which are not necessarily finite dimensional and $\alpha,\beta \in Aut_{Hopf}(H_1), \gamma,\delta \in Aut_{Hopf}(H_2)$. In this paper, we introduce a category ${}_{H_1}\mathcal{LR}_{H_2}(\alpha, \beta,…

We find a formula to compute the number of the generators, which generate the $n$-filtered space of Hopf algebra of rooted trees, i.e. the number of equivalent classes of rooted trees with weight $n$. Applying Hopf algebra of rooted trees,…

For a finitary hereditary abelian category $\mathcal{A}$, we define a derived Hall algebra of its root category by counting the triangles and using the octahedral axiom, which is proved to be isomorphic to the Drinfeld double of Hall…

Fix a finite symmetric tensor category $\mathcal{E}$ over an algebraically closed field. We derive an $\mathcal{E}$-enriched version of Shimizu's characterizations of non-degeneracy for finite braided tensor categories. In order to do so,…

We show that for a braided Hopf algebra in the category of comodules over a cosemisimple coquasitriangular Hopf algebra, the Hochschild cohomological dimension, the left and right global dimensions and the projective dimensions of the…

For a given finite dimensional Hopf algebra $H$ we describe the set of all equivalence classes of cocycle deformations of $H$ as an affine variety, using methods of geometric invariant theory. We show how our results specialize to the…

Twisted tensor powers of quasitriangular Hopf algebras with diagonal sub-Hopf-algebras (self-diagonal tensor powers) are introduced together with their duals and their mutual *-structures as generalizations of the Drinfel'd double as given…

Quantum field theory allows more general symmetries than groups and Lie algebras. For instance quantum groups, that is Hopf algebras, have been familiar to theoretical physicists for a while now. Nowdays many examples of symmetries of…

For a fixed finite group $Q$ and semi-simple finite dimensional algebra $S$, we examine an equivalence between strongly $Q$-graded algebras (extensions) with identity component $S$ and $S^1$-gerbes on action groupoids of $Q$ on the set of…

We explain that a new theorem of Deligne on symmetric tensor categories implies, in a straightforward manner, that any finite dimensional triangular Hopf algebra over an algebraically closed field of characteristic zero has Chevalley…

For a given quasitriangular Hopf algebra $\Ha$ we study relations between the braided group $\tilde \Ha^*$ and Drinfeld's twist. We show that the braided bialgebra structure of $\tilde \Ha^*$ is naturally described by means of twisted…

A natural extension of the Hopf-cyclic cohomology, with coefficients, is introduced to encompass topological Hopf algebras. The topological theory allows to work with infinite dimensional Lie algebras. Furthermore, the category of…

In this paper we show that for an important class of non-trivial Hopf algebras, the Schur indicator is a computable invariant. The Hopf algebras we consider are all abelian extensions; as a special case, they include the Drinfeld double of…

By finite quantum groups we mean Lusztig's finite-dimensional pointed Hopf algebras called quantum Frobenius Kernels [9, 10], and their natural generalizations due to Andruskiewitsch and Schneider [2, 3]. For a Hopf algebra $H$ in a special…

We prove a duality theorem for quantum groupoid (weak Hopf algebra) actions that extends the well-known result for usual Hopf algebras.

We show how to define the tensor product of two braided Hopf algebras.

We prove that the Drinfeld double of the category of sheaves on an orbifold is equivalent to the category of sheves on the corresponding inertia orbifold.

For an action $\alpha$ of a group $G$ on an algebra $R$ (over $\Bbb C$), the crossed product $R\times_\alpha G$ is the vector space of $R$-valued functions with finite support in $G$, together with the twisted convolution product given by…

Let A be an abelian category of finite type and homological dimension 1. Then by results of Green R(A), the extended Hall-Ringel algebra of A, has a natural Hopf algebra structure. We consider its Heisenberg double Heis(A) and study its…