On the Structure of Weak Hopf Algebras

By D. Nikshych. 2001

80 citations recorded by the source corpus.

We study the group of group-like elements of a weak Hopf algebra and derive an analogue of Radford's formula for the fourth power of the antipode S, which implies that the antipode has a finite order modulo, a trivial automorphism. We find a sufficient condition in terms of Tr(S2) for a weak Hopf algebra to be semisimple, discuss relation between semisimplicity and cosemisimplicity, and apply our results to show that a dynamical twisting deformation of a semisimple Hopf algebra is cosemisimple.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1d3fbfc4ce8cc9a3af0753086db9e060d8a0acc2

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Citations in the corpus

Module categories over the Drinfeld double of a finite group (2002, background): “We denote by 1 the trivial cohomology class.”
An analogue of Radford's S4 formula for finite tensor categories (2004, background): “formula we have H ⊗ L⊗B◦ (M ⊗ N ) ∼ (H ⊗ L M) ⊗ L◦ N.”
Tensor categories: a selective guided tour (2008, background): “• We give another example for a purely categorical result that can be proven using weak Hopf algebras: Radford’s formula for S(4) has a generalization to weak Hopf algebras [213], and this can be used to prove that in every fusion category, there exists an isomorphism of tensor functors id → ∗ ∗ ∗∗,”
On fusion categories (2002, unknown): “Furthermore, for any simple object X ∈ C, one has FPdim(X) = √ N , where N is an integer.”
Finite tensor categories (2003, result): “m of tensor functors V∗∗ → L∗ ρ⊗ ∗∗ V⊗L. For Hopf algebras, this follows from Radford’s formula for S4. For weak Hopf algebras, it follows from the Nikshych’s generalization of Radford’s formula, see [N], [ENO]. 2.9. Dimensions of projective objects and degeneracy of the Cartan matrix. The following”