arXiv subjects

Peter Schauenburg

Publications and source records attributed to Peter Schauenburg.

At least 19 recordsLinked to original sources

arXiv2026

Hopf Galois extensions of Hopf algebroids

We study Hopf Galois extensions of Hopf algebroids as a generalization of the theory for Hopf algebras. More precisely, we introduce (skew-)regular comodules and generalize the structure theorem for relative Hopf modules. Also, we show that if $N\subseteq P$ is a left $\mathcal{L}$-Galois extension and $Γ$ is a 2-cocycle of $\mathcal{L}$, then for the twisted comodule algebra ${}_ΓP$, $N\subseteq{}_ΓP$ is a left Hopf Galois extension of the twisted Hopf algebroid $\mathcal{L}^Γ$. We study twisted Drinfeld doubles of Hopf algebroids as examples for the Drinfeld twist theory. Finally, we introduce cleft extension and $σ$-twisted crossed products of Hopf algebroids. Moreover, we show the equivalence of cleft extensions, $σ$-twisted crossed products, and Hopf Galois extensions with normal basis properties, which generalize the theory of cleft extensions of Hopf algebras.

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arXiv2026

Cleft Extensions for Hopf Algebroids without Antipodes

We introduce cleft extensions for Hopf algebroids. We prove the equivalence between cleft extensions, $σ$-twisted crossed products, and Hopf-Galois extensions with the normal basis property, thereby generalizing the theory of cleft extensions for Hopf algebroids developed by B{ö}hm and Brzezi{ń}ski, and fitting in with the general theory of Galois and biGalois extensions over Hopf algebroids developed by the authors. We investigate the Ehresmann Hopf algebroid associated with a cleft extension and show that it is isomorphic to a generalized version of the Connes-Moscovici Hopf algebroid. A special case of the Connes-Moscovici Hopf algebroid, namely the case where the coinvariants of the cleft extension coincide with the base of the Hopf algebroid, is a Drinfeld twist of a Hopf algebroid by a two-cocycle, generalizing work of B{ö}hm, Han and Majid.

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arXiv2026

Hopf BiGalois Theory for Hopf Algebroids

We develop a theory of Hopf BiGalois extensions for Hopf algebroids. We understand these to be left bialgebroids (whose left module categories are monoidal categories) fulfilling a condition that is equivalent to being Hopf in the case of ordinary bialgebras, but does not entail the existence of an antipode map. The immediate obstacle to developing a full biGalois theory for such Hopf algebroids is simple: The condition to be a left Hopf Galois extension can be defined in complete analogy to the Hopf case, but the Galois map for a right comodule algebra is not a well defined map. We find that this obstacle can be circumvented using bialgebroids fulfilling a condition that still does not entail the existence of an antipode, but is equivalent, for ordinary bialgebras, to being Hopf with bijective antipode. The key technical tool is a result of Chemla allowing to switch left and right comodule structures under flatness conditions much like one would do using an antipode. Using this, we arrive at a left-right symmetric theory of biGalois extensions, including the construction of an Ehresmann Hopf algebroid making a one-sided Hopf-Galois extension into a biGalois one. Moreover, we apply a more general 2-cocycle twist theory to Ehresmann Hopf algebroids. As the 2-cocycle is only left linear over the base, the base algebra is also twisted. We also study the Ehresmann Hopf algebroids of quantum Hopf fibrations and quantum homogeneous space as examples.

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arXiv2025

Reflective centers as categories of modules

In [LWY23] the authors construct the reflective center of a module category M over a braided monoidal category B. The reflective center is by construction a braided module category over B. In the case where B is the category of modules over a finite dimensional quasitriangular Hopf algebra H, acting on the category of modules over a comodule algebra, they construct a comodule algebra, the reflective algebra, whose modules are precisely the reflective center. In the construction, Majid's transmutation of H plays a crucial r{ô}le. This note centers on the transmuted H, seeking to ''explain'' its appearance through a generalization in which the acting category is no longer a module category, but admits an internal reconstructed Hopf algebra; the transmutation is a special case of this notion. As a result, in certain cases, the reflective center is simply the category of modules in M over that Hopf algebra in B.

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arXiv2021

Modular categories are not determined by their modular data

Arbitrarily many pairwise inequivalent modular categories can share the same modular data. We exhibit a family of examples that are module categories over twisted Drinfeld doubles of finite groups, and thus in particular integral modular categories.

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arXiv2021

A Topological Invariant for Modular Fusion Categories

The modular data of a modular category $\mathcal{C}$, consisting of the $S$-matrix and the $T$-matrix, is known to be an incomplete invariant of $\mathcal{C}$. More generally, the invariants of framed links and knots defined by a modular category as part of a topological quantum field theory can be viewed as numerical invariants of the category. Among these invariants, we study the invariant defined by the Borromean link colored by three objects. Thus we obtain a tensor that we call $B$. We derive a formula for the Borromean tensor for the twisted Drinfeld doubles of finite groups. Along with $T$, it distinguishes the $p$ non-equivalent modular categories of the form $\mathcal{Z}({\rm Vec}_G^ω)$ for $G$ the non-abelian group $\mathbb{Z}/q\mathbb{Z} \rtimes \mathbb{Z}/p\mathbb{Z}$, which are not distinguished by the modular data.

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arXiv2017

Morita equivalence of pointed fusion categories of small rank

We classify pointed fusion categories C(G, $ω$) up to Morita equivalence for 1 < |G| < 32. Among them, the cases |G| = 2 3 , 2 4 and 3 3 are emphasized. Although the equivalence classes of such categories are not distinguished by their Frobenius-Schur indicators, their categorical Morita equivalence classes are distinguished by the set of the indicators and ribbon twists of their Drinfeld centers. In particular, the modular data are a complete invariant for the modular categories Z(C(G, $ω$)) for |G[< 32. We use the computer algebra package GAP and present codes for treating complex-valued group cohomology and calculating Frobenius-Schur indicators.

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arXiv2016

Higher frobenius-schur indicators for drinfeld doubles of finite groups through characters of centralizers

We present a new approach to calculating the higher Frobenius-Schur indicators for the simple modules over the Drinfeld double of a finite group. In contrast to the formula by Kashina-Sommerh{ä}user-Zhu that involves a sum over all group elements satisfying a certain condition, our formula operates on the level of conjugacy classes and character tables. It can be implemented in the computer algebra system GAP, efficiently enough to deal, on a laptop, with symmetric groups up to $S_{18}$ (providing further evidence that indicators are non-negative in this case) or simple groups of order up to $2 \cdot 10^8$. The approach also allows us to test whether all indicators over the double of a given group are rational , without computing them. Among simple groups of order up to about $5 \cdot 10^{11}$ an inspection yields exactly one example (of order about $5 \cdot 10^9$) where irrational indicators occur.

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arXiv2015

A note on the restricted universal enveloping algebra of a restricted Lie-Rinehart Algebra

Lie-Rinehart algebras, also known as Lie algebroids, give rise to Hopf algebroids by a universal enveloping algebra construction, much as the universal enveloping algebra of an ordinary Lie algebra gives a Hopf algebra, of infinite dimension. In finite characteristic, the universal enveloping algebra of a restricted Lie algebra admits a quotient Hopf algebra which is finite-dimensional if the Lie algebra is. Rumynin has shown that suitably defined restricted Lie algebroids allow to define restricted universal enveloping algebras that are finitely generated projective if the Lie algebroid is. This note presents an alternative proof and possibly fills a gap that might, however, only be a gap in the author's understanding.

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arXiv2015

The dual and the double of a Hopf algebroid are Hopf algebroids

Let $H$ be a $\times$-bialgebra in the sense of Takeuchi. We show that if $H$ is $\times$-Hopf, and if $H$ fulfills the finiteness condition necessary to define its skew dual $H^\vee$, then the coopposite of the latter is $\times$-Hopf as well. If in addition the coopposite $\times$-bialgebra of $H$ is $\times$-Hopf, then the coopposite of the Drinfeld double of $H$ is $\times$-Hopf, as is the Drinfeld double itself, under an additional finiteness condition.

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arXiv2015

Frobenius-Schur indicators for some fusion categories associated to symmetric and alternating groups

We calculate Frobenius-Schur indicator values for some fusion categories obtained from inclusions of finite groups $H\subset G$, where more concretely $G$ is symmetric or alternating, and $H$ is a symmetric, alternating or cyclic group. Our work is strongly related to earlier results by Kashina-Mason-Montgomery, Jedwab-Montgomery, and Timmer for bismash product Hopf algebras obtained from exact factorizations of groups. We can generalize some of their results, settle some open questions and offer shorter proofs; this already pertains to the Hopf algebra case, while our results also cover fusion categories not associated to Hopf algebras.

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arXiv2015

Some quasitensor autoequivalences of Drinfeld doubles of finite groups

We report on two classes of autoequivalences of the category of Yetter-Drinfeld modules over a finite group, or, equivalently the Drinfeld center of the category of representations of a finite group. Both operations are related to the $r$-th power operation, with $r$ relatively prime to the exponent of the group. One is defined more generally for the group-theoretical fusion category defined by a finite group and an arbitrary subgroup, while the other seems particular to the case of Yetter-Drinfeld modules. Both autoequivalences preserve higher Frobenius-Schur indicators up to Galois conjugation, and they preserve tensor products, although neither of them can in general be endowed with the structure of a monoidal functor.

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arXiv2015

A Higher Frobenius-Schur Indicator Formula for Group-Theoretical Fusion Categories

Group-theoretical fusion categories are defined by data concerning finite groups and their cohomology: A finite group $G$ endowed with a three-cocycle $ω$, and a subgroup $H\subset G$ endowed with a two-cochain whose coboundary is the restriction of $ω$. The objects of the category are $G$-graded vector spaces with suitably twisted $H$-actions; the associativity of tensor products is controlled by $ω$. Simple objects are parametrized in terms of projective representations of finite groups, namely of the stabilizers in $H$ of right $H$-cosets in $G$, with respect to two-cocycles defined by the initial data. We derive and study general formulas that express the higher Frobenius-Schur indicators of simple objects in a group-theoretical fusion category in terms of the group-theoretical and cohomological data defining the category and describing its simples.

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arXiv2015

Computing Higher Frobenius-Schur Indicators in Fusion Categories Constructed from Inclusions of Finite Groups

We consider a subclass of the class of group-theoretical fusion categories: To every finite group $G$ and subgroup $H$ one can associate the category of $G$-graded vector spaces with a two-sided $H$-action compatible with the grading. We derive a formula that computes higher Frobenius-Schur indicators for the objects in such a category using the combinatorics and representation theory of the groups involved in their construction. We calculate some explicit examples for inclusions of symmetric groups.

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arXiv2014

On tensor factorizations of Hopf algebras

We prove a variety results on tensor product factorizations of finite dimensional Hopf algebras (more generally Hopf algebras satisfying chain conditions in suitable braided categories). The results are analogs of well-known results on direct product factorizations of finite groups (or groups with chain conditions) such as Fitting's Lemma and the uniqueness of the Krull-Remak-Schmidt factorization. We analyze the notion of normal (and conormal) Hopf algebra endomorphisms, and the structure of endomorphisms and automorphisms of tensor products. The results are then applied to compute the automorphism group of the Drinfeld double of a finite group in the case where the group contains an abelian factor. (If it doesn't, the group can be calculated by results of the first author.)

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arXiv2010

Congruence Subgroups and Generalized Frobenius-Schur Indicators

We introduce generalized Frobenius-Schur indicators for pivotal categories. In a spherical fusion category C, an equivariant indicator of an object in C is defined as a functional on the Grothendieck algebra of the quantum double Z(C) via generalized Frobenius-Schur indicators. The set of all equivariant indicators admits a natural action of the modular group. Using the properties of equivariant indicators, we prove a congruence subgroup theorem for modular categories. As a consequence, all modular representations of a modular category have finite images, and they satisfy a conjecture of Eholzer. In addition, we obtain two formulae for the generalized indicators, one of them a generalization of Bantay's second indicator formula for a rational conformal field theory. This formula implies a conjecture of Pradisi-Sagnotti-Stanev, as well as a conjecture of Borisov-Halpern-Schweigert.

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arXiv2006

Frobenius-Schur Indicators and Exponents of Spherical Categories

We obtain two formulae for the higher Frobenius-Schur indicators: one for a spherical fusion category in terms of the twist of its center and the other one for a modular tensor category in terms of its twist. The first one is a categorical generalization of an analogous result by Kashina, Sommerhauser, and Zhu for Hopf algebras, and the second one extends Bantay's 2nd indicator formula for a conformal field theory to higher degree. These formulae imply the sequence of higher indicators of an object in these categories is periodic. We define the notion of Frobenius-Schur (FS-)exponent of a pivotal category to be the global period of all these sequences of higher indicators, and we prove that the FS-exponent of a spherical fusion category is equal to the order of the twist of its center. Consequently, the FS-exponent of a spherical fusion category is a multiple of its exponent, in the sense of Etingof, by a factor not greater than 2. As applications of these results, we prove that the exponent and the dimension of a semisimple quasi-Hopf algebra H have the same prime divisors, which answers two questions of Etingof and Gelaki affirmatively for quasi-Hopf algebras. Moreover, we prove that the FS-exponent of H divides dim(H)^4. In addition, if H is a group-theoretic quasi-Hopf algebra, the FS-exponent of H divides dim(H)^2, and this upper bound is shown to be tight.

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arXiv2005

Higher Frobenius-Schur Indicators for Pivotal Categories

We define higher Frobenius-Schur indicators for objects in linear pivotal monoidal categories. We prove that they are category invariants, and take values in the cyclotomic integers. We also define a family of natural endomorphisms of the identity endofunctor on a $k$-linear semisimple rigid monoidal category, which we call the Frobenius-Schur endomorphisms. For a $k$-linear semisimple pivotal monoidal category -- where both notions are defined --, the Frobenius-Schur indicators can be computed as traces of the Frobenius-Schur endomorphisms.

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