225 research outputs found

### Hopf algebras of prime dimension in positive characteristic

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

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

### Central Invariants and Higher Indicators for Semisimple Quasi-Hopf Algebras

In this paper, we define the higher Frobenius-Schur (FS-)indicators for
finite-dimensional modules $V$ of a semisimple quasi-Hopf algebra $H$ 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 $H$ is a Hopf algebra. We also obtain a
sequence of canonical central elements of $H$, which is invariant under gauge
transformations, whose values, when evaluated by the character of an $H$-module
$V$, are the higher Frobenius-Schur indicators of $V$. 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

### Hopf algebras of dimension pq

Let H be a non-semisimple Hopf algebra with antipode S of dimension pq over an algebraically closed field of characteristic 0 where p β€ q are odd primes. We prove that Tr(S2p) = p2d where d β‘ 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

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

The definitions of the $n^{th}$ Gauss sum and the associated $n^{th}$ central
charge are introduced for premodular categories $\mathcal{C}$ and
$n\in\mathbb{Z}$. We first derive an expression of the $n^{th}$ Gauss sum of a
modular category $\mathcal{C}$, for any integer $n$ coprime to the order of the
T-matrix of $\mathcal{C}$, in terms of the first Gauss sum, the global
dimension, the twist and their Galois conjugates. As a consequence, we show for
these $n$, the higher Gauss sums are $d$-numbers and the associated central
charges are roots of unity. In particular, if $\mathcal{C}$ 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 $p^n$ in characteristic $p$

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

- β¦