Classification of semisimple Hopf algebras of dimension 16

In this paper we completely classify nontrivial semisimple Hopf algebras of dimension 16. We also compute all the possible structures of the Grothendieck ring of semisimple non-commutative Hopf algebras of dimension 16. Moreover, we prove that non-commutative semisimple Hopf algebras of dimension p^n, p is prime, cannot have a cyclic group of grouplikes.Comment: 31 page

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.Comment: 62 pages. Important new result added, remark by P. Etingof included, mistake in last section corrected. See also http://www.mathematik.uni-muenchen.de/~sommer

Computing the Frobenius-Schur indicator for abelian extensions of Hopf algebras

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 a group algebra. In addition to finding a general formula for the indicator, we also study when it is always positive. In particular we prove that the indicator is always positive for the Drinfeld double of the symmetric group, generalizing the classical result for the symmetric group itself.Comment: 22 pages; the main result of Section 3 has been corrected in the revised version. The formula in Theorem 4.4 has also been affected by this correctio

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.Comment: 9 pages. Important new result included. See also http://www.mathematik.uni-muenchen.de/~sommer

On Isomorphisms between Certain Yetter-Drinfel'd Hopf Algebras

For two families of Yetter-Drinfel'd Hopf algebras considered earlier by the authors, we determine which of them are isomorphic. We also determine which of their Radford biproducts are isomorphic.Comment: 23 pages. In the second version, the argument in Paragraph 1.5 has been simplified. See also https://www.math.mun.ca/~sommerh

On the trace of the antipode and higher indicators

We introduce two kinds of gauge invariants for any finite-dimensional Hopf algebra H. When H is semisimple over C, these invariants are, respectively, the trace of the map induced by the antipode on the endomorphism ring of a self-dual simple module, and the higher Frobenius-Schur indicators of the regular representation. We further study the values of these higher indicators in the context of complex semisimple quasi-Hopf algebras H. We prove that these indicators are non-negative provided the module category over H is modular, and that for a prime p, the p-th indicator is equal to 1 if, and only if, p is a factor of dimH. As an application, we show the existence of a non-trivial self-dual simple H-module with bounded dimension which is determined by the value of the second indicator. © 2011 Hebrew University Magnes Press

Classification of integral modular categories of Frobenius-Perron dimension pq^4 and p^2q^2

We classify integral modular categories of dimension pq^4 and p^2q^2 where p and q are distinct primes. We show that such categories are always group-theoretical except for categories of dimension 4q^2. In these cases there are well-known examples of non-group-theoretical categories, coming from centers of Tambara-Yamagami categories and quantum groups. We show that a non-group-theoretical integral modular category of dimension 4q^2 is equivalent to either one of these well-known examples or is of dimension 36 and is twist-equivalent to fusion categories arising from a certain quantum group.Comment: 13 page

Classification of integral modular categories of Frobenius–Perron dimension pq4 and p2q2

We classify integral modular categories of dimension pq4 and p2q2, where p and q are distinct primes. We show that such categories are always group-theoretical, except for categories of dimension 4q2. In these cases there are well-known examples of non-group-theoretical categories, coming from centers of Tambara–Yamagami categories and quantum groups. We show that a non-group-theoretical integral modular category of dimension 4q2 is either equivalent to one of these well-known examples or is of dimension 36 and is twist-equivalent to fusion categories arising from a certain quantum group.submittedVersionFil: Bruillard, Paul. Texas A&M University. Department of Mathematics; United States of America.Fil: Hong, Seung-Moon. University of Toledo. Department of Mathematics and Statistics; United States of America.Fil: Kashina, Yevgenia. DePaul University. Department of Mathematical Sciences; United States of America.Fil: Naidu, Deepak. Northern Illinois University. Department of Mathematical Sciences; United States of America.Fil: Rowell, Eric C. Texas A&M University. Department of Mathematics; United States of America.Fil: Galindo Martínez, César Neyit. Universidad de los Andes. Facultad de Ciencias. Departamento de Matemáticas; Colombia.Fil: Natale, Sonia Luján. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Fil: Natale, Sonia Luján. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina.Fil: Natale, Sonia Luján. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina.Fil: Plavnik, Julia Yael. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Fil: Plavnik, Julia Yael. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina.Fil: Plavnik, Julia Yael. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina.Matemática Pur