A Note on Frobenius-schur Indicators
ثبت نشده
چکیده
This exposition concerns two different notions of Frobenius-Schur indicators for finite-dimensional Hopf algebras. These two versions of indicators coincide when the underlying Hopf algebra is semisimple. We are particularly interested in the family of pivotal finite-dimensional Hopf algebras with unique pivotal element; both indicators are gauge invariants of this family of Hopf algebras. We obtain a formula for the (pivotal) Frobenius-Schur indicators for the regular representation of a pivotal Hopf algebra. In particular, we use this formula for the four dimensional Sweedler algebra and demonstrate the difference of these two indicators.
منابع مشابه
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 ≥ 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. Bu...
متن کاملDuality, Central Characters, and Real-valued Characters of Finite Groups of Lie Type
We prove that the duality operator preserves the Frobenius–Schur indicators of characters of connected reductive groups of Lie type with connected center. This allows us to extend a result of D. Prasad which relates the Frobenius–Schur indicator of a regular real-valued character to its central character. We apply these results to compute the Frobenius–Schur indicators of certain real-valued, i...
متن کاملCongruence Subgroups and Generalized Frobenius-schur Indicators
We define generalized Frobenius-Schur indicators for objects in a linear pivotal category C. An equivariant indicator of an object is defined as a functional on the Grothendieck algebra of the quantum double Z(C) of C using the values of the generalized Frobenius-Schur indicators. In a spherical fusion category C with Frobenius-Schur exponent N , we prove that the set of all equivariant indicat...
متن کاملTwisted Exponents and Twisted Frobenius–schur Indicators for Hopf Algebras
Classically, the exponent of a group is the least common multiple of the orders of its elements. This notion was generalized by Etingof and Gelaki to the context of Hopf algebras. Kashina, Sommerhäuser and Zhu later observed that there is a strong connection between exponents and Frobenius– Schur indicators. In this paper, we introduce the notion of twisted exponents and show that there is a si...
متن کاملFrobenius-schur Indicators for a Class of Fusion Categories
We give an explicit description, up to gauge equivalence, of group-theoretical quasi-Hopf algebras. We use this description to compute the Frobenius-Schur indicators for grouptheoretical fusion categories.
متن کامل