271 research outputs found
Frobenius-Schur indicator for categories with duality
We introduce the Frobenius-Schur indicator for categories with duality to
give a category-theoretical understanding of various generalizations of the
Frobenius-Schur theorem, including that for semisimple quasi-Hopf algebras,
weak Hopf C*-algebras and association schemes. Our framework also clarifies a
mechanism how the `twisted' theory arises from the ordinary case. As a
demonstration, we give a twisted Frobenius-Schur theorem for semisimple
quasi-Hopf algebras. We also give several applications to the quantum SL_2.Comment: 38 pages; final version published in the Special Issue on "Hopf
Algebras, Quantum Groups and Yang-Baxter Equations" of Axiom
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 -linear semisimple rigid
monoidal category, which we call the Frobenius-Schur endomorphisms. For a
-linear semisimple pivotal monoidal category -- where both notions are
defined --, the Frobenius-Schur indicators can be computed as traces of the
Frobenius-Schur endomorphisms.Comment: A paragraph which describes the organization of the paper has been
added to the introduction. Some observations have been added to Theorems 5.1
and 7.
A reason for fusion rules to be even
We show that certain tensor product multiplicities in semisimple braided
sovereign tensor categories must be even. The quantity governing this behavior
is the Frobenius-Schur indicator. The result applies in particular to the
representation categories of large classes of groups, Lie algebras, Hopf
algebras and vertex algebras.Comment: 6 pages, LaTe
The pivotal cover and Frobenius-Schur indicators
In this paper, we introduce the notion of the pivotal cover
of a left rigid monoidal category to
develop a theoretical foundation for the theory of Frobenius-Schur (FS)
indicators in "non-pivotal" settings. For an object , the -th FS indicator is defined by generalizing that of an object of a pivotal
monoidal category. This notion gives a categorical viewpoint to some recent
results on generalizations of FS indicators.
Based on our framework, we also study the FS indicators of the "adjoint
object" in a finite tensor category, which can be considered as a
generalization of the adjoint representation of a Hopf algebra. The indicators
of this object closely relate to the space of endomorphisms of the iterated
tensor product functor.Comment: The final version accepted for publication in Journal of Algebra (37
pages, many figures
Twisted Frobenius-Schur indicators for Hopf algebras
The classical Frobenius-Schur indicators for finite groups are character sums
defined for any representation and any integer m greater or equal to 2. In the
familiar case m=2, the Frobenius-Schur indicator partitions the irreducible
representations over the complex numbers into real, complex, and quaternionic
representations. In recent years, several generalizations of these invariants
have been introduced. Bump and Ginzburg, building on earlier work of Mackey,
have defined versions of these indicators which are twisted by an automorphism
of the group. In another direction, Linchenko and Montgomery have defined
Frobenius-Schur indicators for semisimple Hopf algebras. In this paper, the
authors construct twisted Frobenius-Schur indicators for semisimple Hopf
algebras; these include all of the above indicators as special cases and have
similar properties.Comment: 12 pages. Minor revision
Frobenius-Schur indicators in Tambara-Yamagami categories
We introduce formulae of Frobenius-Schur indicators of simple objects of
Tambara-Yamagami categories. By using techniques of the Fourier transform on
finite abelian groups, we study some arithmetic properties of indicators.Comment: 21 page
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