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Yorck Sommerhaeuser

Publications and source records attributed to Yorck Sommerhaeuser.

14 recordsLinked to original sources
arXiv2026

Hochschild Cohomology, Modular Tensor Categories, and Mapping Class Groups II

In the first part of this work, we have, for a not necessarily semisimple modular category, generalized the action of the mapping class groups of surfaces on the spaces of conformal blocks to the so-called derived block spaces. In the second part presented here, we compute this action explicitly in the case of Drinfel'd doubles of finite groups over fields of positive characteristic. To do that, we connect Lyubashenko's approach to mapping class group representations with the theory of representation varieties. In this way, we are able to show that the mapping class group representations on the derived block spaces are in general different from those on the ordinary block spaces.

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arXiv2021

Hochschild Cohomology, Modular Tensor Categories, and Mapping Class Groups

Given a finite modular tensor category, we associate with each compact surface with boundary a cochain complex in such a way that the mapping class group of the surface acts projectively on its cohomology groups. In degree zero, this action coincides with the known projective action of the mapping class group on the space of chiral conformal blocks. In the case that the surface is a torus and the category is the representation category of a factorizable ribbon Hopf algebra, we recover our previous result on the projective action of the modular group on the Hochschild cohomology groups of the Hopf algebra.

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arXiv2020

On Biproducts and Extensions

We describe in which ways the Radford biproducts of certain eight-dimensional Yetter-Drinfel'd Hopf algebras over the elementary abelian group of order 4 can be written as extensions of Hopf algebras.

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arXiv2019

On Cores in Yetter-Drinfel'd Hopf Algebras

By constructing explicit examples, we show that the core of a group-like element in a cocommutative cosemisimple Yetter-Drinfel'd Hopf algebra over the group ring of a finite abelian group is not always completely trivial.

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arXiv2017

Hochschild Cohomology and the Modular Group

It has been shown in previous work that the modular group acts projectively on the center of a factorizable ribbon Hopf algebra. The center is the zeroth Hochschild cohomology group. In this article, we extend this projective action of the modular group to an arbitrary Hochschild cohomology group of a factorizable ribbon Hopf algebra, in fact up to homotopy even to a projective action on the entire Hochschild cochain complex.

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arXiv2016

Triviality Theorems for Yetter-Drinfel'd Hopf Algebras

Under suitable assumptions on the base field, we prove that a commutative semisimple Yetter-Drinfel'd Hopf algebra over a finite abelian group is trivial, i.e., is an ordinary Hopf algebra, if its dimension is relatively prime to the order of the finite abelian group. Furthermore, we prove that a finite-dimensional cocommutative cosemisimple Yetter-Drinfel'd Hopf algebra contains a trivial Yetter-Drinfel'd Hopf subalgebra of dimension greater than one, at least if the Yetter-Drinfel'd Hopf algebra itself has dimension greater than one.

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arXiv2009

On the Notion of a Ribbon Quasi-Hopf Algebra

We show that two competing definitions of a ribbon quasi-Hopf algebra are actually equivalent. Along the way, we look at the Drinfel'd element from a new perspective and use this viewpoint to derive its fundamental properties.

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arXiv2009

On the Central Charge of a Factorizable Hopf Algebra

For a semisimple factorizable Hopf algebra over a field of characteristic zero, we show that the value that an integral takes on the inverse Drinfel'd element differs from the value that it takes on the Drinfel'd element itself at most by a fourth root of unity. This can be reformulated by saying that the central charge of the Hopf algebra is an integer. If the dimension of the Hopf algebra is odd, we show that these two values differ at most by a sign, which can be reformulated by saying that the central charge is even. We give a precise condition on the dimension that determines whether the plus sign or the minus sign occurs. To formulate our results, we use the language of modular data.

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arXiv2008

Hopf Algebras and Congruence Subgroups

We prove that the kernel of the natural action of the modular group on the center of the Drinfel'd double of a semisimple Hopf algebra is a congruence subgroup. To do this, we introduce a class of generalized Frobenius-Schur indicators and endow it with an action of the modular group that is compatible with the original one.

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arXiv2005

On Higher Frobenius-Schur Indicators

We study the higher Frobenius-Schur indicators of modules over semisimple Hopf algebras, and relate them to other invariants as the exponent, the order, and the index. We prove various divisibility and integrality results for these invariants. In particular, we prove a version of Cauchy's theorem for semisimple Hopf algebras. Furthermore, we give some examples that illustrate the general theory.

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arXiv2004

Stable anti-Yetter-Drinfeld modules

We define and study a class of entwined modules (stable anti-Yetter-Drinfeld modules) that serve as coefficients for the Hopf-cyclic homology and cohomology. In particular, we explain their relationship with Yetter-Drinfeld modules and Drinfeld doubles. Among sources of examples of stable anti-Yetter-Drinfeld modules, we find Hopf-Galois extensions with a flipped version of the Miyashita-Ulbrich action.

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arXiv2001

Self-dual modules of semisimple Hopf algebras

We prove that, over an algebraically closed field of characteristic zero, a semisimple Hopf algebra that has a nontrivial self-dual simple module must have even dimension. This generalizes a classical result of W. Burnside. As an application, we show under the same assumptions that a semisimple Hopf algebra that has a simple module of even dimension must itself have even dimension.

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arXiv1999

Yetter-Drinfel'd Hopf algebras over groups of prime order

We prove a structure theorem for Yetter-Drinfel'd Hopf algebras over groups of prime order that are nontrivial, cocommutative, and cosemisimple: Under certain assumptions on the base field, these algebras can be decomposed into a tensor product of the dual group ring of the group of prime order and an ordinary group ring of some other group. This tensor product is a crossed product as an algebra and an ordinary tensor product as a coalgebra. In particular, the dimension of such a Yetter-Drinfel'd Hopf algebra is divisible by the prime under consideration. We also find explicit examples of such Yetter-Drinfel'd Hopf algebras and apply the result to the classification program for semisimple Hopf algebras.

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