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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, Sommerhauser 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 similar relationship between the twisted exponent and the twisted Frobenius-Schur indicators defined in previous work of the authors. In particular, we exhibit a new formula for the twisted Frobenius-Schur indicators and use it to prove periodicity and rationality statements for the twisted indicators.
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 o
Let g be a finite dimensional complex semisimple Lie algebra, and let V be a finite dimensional represenation of g. We give a closed formula for the mth Frobenius-Schur indicator, m>1, of V in representation-theoretic terms. We deduce that the indica
We define total Frobenius-Schur indicator for each object in a spherical fusion category $C$ as a certain canonical sum of its higher indicators. The total indicators are invariants of spherical fusion categories. If $C$ is the representation categor
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
A representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral Z_2-lattice. The irreducible decomposition of the representation is explicitly