225 research outputs found

    Hopf algebras of prime dimension in positive characteristic

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    We prove that a Hopf algebra of prime dimension pp over an algebraically closed field, whose characteristic is equal to pp, is either a group algebra or a restricted universal enveloping algebra. Moreover, we show that any Hopf algebra of prime dimension pp over a field of characteristic q>0q>0 is commutative and cocommutative when q=2q=2 or p<4qp<4q. This problem remains open in positive characteristic when 2<q<p/42<q<p/4.Comment: 7 pages; to appear in Bulletin of the London Mathematical Societ

    Central Invariants and Higher Indicators for Semisimple Quasi-Hopf Algebras

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    In this paper, we define the higher Frobenius-Schur (FS-)indicators for finite-dimensional modules VV of a semisimple quasi-Hopf algebra HH via the categorical counterpart developed in \cite{NS05}. We prove that this definition of higher FS-indicators coincides with the higher indicators introduced by Kashina, Sommerh\"auser, and Zhu when HH is a Hopf algebra. We also obtain a sequence of canonical central elements of HH, which is invariant under gauge transformations, whose values, when evaluated by the character of an HH-module VV, are the higher Frobenius-Schur indicators of VV. As an application, we show that FS-indicators are sufficient to distinguish the four gauge equivalence classes of semisimple quasi-Hopf algebras of dimension eight corresponding to the four fusion categories with certain fusion rules classified by Tambara and Yamagami. Three of these categories correspond to well-known Hopf algebras, and we explicitly construct a quasi-Hopf algebra corresponding to the fourth. We also derive explicit formulae for FS-indicators for some quasi-Hopf algebras associated to group cocycles.Comment: The higher Frobenius-Schur indicators for certain quasi-Hopf algebras associated with finite groups and their 3-cocycles have been computed in section

    Higher Frobenius-Schur Indicators for Pivotal Categories

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    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 kk-linear semisimple rigid monoidal category, which we call the Frobenius-Schur endomorphisms. For a kk-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.

    Hopf algebras of dimension pq

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    Let H be a non-semisimple Hopf algebra with antipode S of dimension pq over an algebraically closed field of characteristic 0 where p ≀ q are odd primes. We prove that Tr(S2p) = p2d where d ≑ pq (mod 4). As a consequence, if p, q are twin primes, then any Hopf algebra of dimension pq is semisimple. Β© 2004 Elsevier Inc. All rights reserved

    Congruence Property In Conformal Field Theory

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    The congruence subgroup property is established for the modular representations associated to any modular tensor category. This result is used to prove that the kernel of the representation of the modular group on the conformal blocks of any rational, C_2-cofinite vertex operator algebra is a congruence subgroup. In particular, the q-character of each irreducible module is a modular function on the same congruence subgroup. The Galois symmetry of the modular representations is obtained and the order of the anomaly for those modular categories satisfying some integrality conditions is determined.Comment: References are updated. Some typos and grammatical errors are correcte

    Higher Gauss sums of modular categories

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    The definitions of the nthn^{th} Gauss sum and the associated nthn^{th} central charge are introduced for premodular categories C\mathcal{C} and n∈Zn\in\mathbb{Z}. We first derive an expression of the nthn^{th} Gauss sum of a modular category C\mathcal{C}, for any integer nn coprime to the order of the T-matrix of C\mathcal{C}, in terms of the first Gauss sum, the global dimension, the twist and their Galois conjugates. As a consequence, we show for these nn, the higher Gauss sums are dd-numbers and the associated central charges are roots of unity. In particular, if C\mathcal{C} is the Drinfeld center of a spherical fusion category, then these higher central charges are 1. We obtain another expression of higher Gauss sums for de-equivariantization and local module constructions of appropriate premodular and modular categories. These expressions are then applied to prove the Witt invariance of higher central charges for pseudounitary modular categories.Comment: 26 pages. Typos and minor mistakes are corrected. Question 7.3 in the previous version is answere

    On Hopf algebras of dimension pnp^n in characteristic pp

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    Let k\Bbbk be an algebraically closed field of characteristic p>0p>0. We study the general structures of pnp^n-dimensional Hopf algebras over k\Bbbk with pnβˆ’1p^{n-1} group-like elements or a primitive element generating a pnβˆ’1p^{n-1}-dimensional Hopf subalgebra. As applications, we have proved that Hopf algebras of dimension p2p^2 over k\Bbbk are pointed or basic for p≀5p \le 5, and provided a list of characterizations of the Radford algebra R(p)R(p). In particular, R(p)R(p) is the unique nontrivial extension of k[Cp]βˆ—\Bbbk[C_p]^* by k[Cp]\Bbbk[C_p], where CpC_p is the cyclic group of order pp. In addition, we have proved a vanishing theorem for some 2nd Sweedler cohomology group and investigated the extensions of pp-dimensional Hopf algebras. All these extensions have been identified and shown to be pointed.Comment: 31 pages LaTe
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