Higher Frobenius-Schur indicators for pivotal categories

Ng, Siu-Hung; Schauenburg, Peter

doi:10.1090/conm/441/08500

Cited by 83 publications

(129 citation statements)

References 14 publications

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“…As shown in [Ng and Schauenburg 2007], F S p (V ) does not depend on the choice of f , and we have F S p (V ) = ±1 (or 0).…”

Section: B the Frobenius-schur Indicatormentioning

confidence: 58%

“…The following definition of the Frobenius-Schur indicator for a pivotal category was given by [Ng and Schauenburg 2007] (see also [Linchenko and Montgomery 2000] for the case of Hopf algebras and [Fuchs et al 1999] for C * sovereign categories).…”

Section: B the Frobenius-schur Indicatormentioning

confidence: 99%

“…Notice that the Frobenius-Schur indicator depends on the pivotal structure, which in turn depends on the choice of ribbon element. As described in [Ng and Schauenburg 2007], we use F S v (V ) to denote the Frobenius-Schur indicator for V , calculated with the ribbon element v.…”

Section: B the Frobenius-schur Indicatormentioning

confidence: 99%

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The half-twist forU_q(g) representations

Snyder

Tingley

2009

ANT

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We introduce the notion of a half-ribbon Hopf algebra, which is a Hopf algebra H along with a distinguished element t ∈ H such that (H, R, C) is a ribbon Hopf algebra, where R = (t −1 ⊗ t −1 ) (t) and C = t −2 . The element t is closely related to the topological "half-twist", which twists a ribbon by 180 degrees. We construct a functor from a topological category of ribbons with half-twists to the category of representations of any half-ribbon Hopf algebra. We show that U q (g) is a (topological) half-ribbon Hopf algebra, but that t −2 is not the standard ribbon element. For U q (sl 2 ), we show that there is no half-ribbon element t such that t −2 is the standard ribbon element. We then discuss how ribbon elements can be modified, and some consequences of these modifications.

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“…As shown in [Ng and Schauenburg 2007], F S p (V ) does not depend on the choice of f , and we have F S p (V ) = ±1 (or 0).…”

Section: B the Frobenius-schur Indicatormentioning

confidence: 58%

Section: B the Frobenius-schur Indicatormentioning

confidence: 99%

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The half-twist forU_q(g) representations

Snyder

Tingley

2009

ANT

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“…Indeed, by the coherence theorems [17] every monoidal category is equivalent to a strict one, and this extends to categories with additional structure, see e.g. [21]. In most applications the monoidal category of interest can safely be replaced with the equivalent strict monoidal category.…”

Section: Graphical Calculusmentioning

confidence: 99%

The graphical calculus for ribbon categories: Algebras, modules, Nakayama automorphisms

FUCHS¹

2006

JNMP

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The graphical description of morphisms in rigid monoidal categories, in particular in ribbon categories, is summarized. It is illustrated with various examples of algebraic structures in such categories, like algebras, (weak) bi-algebras, Frobenius algebras, and modules and bimodules. Nakayama automorphisms of Frobenius algebras are introduced; they are inner iff the algebra is symmetric. Algebras in monoidal categoriesA (unital, associative) algebra is a triple A = (Ȧ, m, η) consisting of a vecor spaceȦ over some field (or more generally, commutative ring) k, a bilinear map m :Ȧ ×Ȧ →Ȧ and an element e ∈Ȧ such that the associativity and unit propertiesfor all a, b, c ∈Ȧ and m(e, a) = a = m(a, e) for all a ∈Ȧ (1.1)hold. The datum η in the triple A is the linear map from k toȦ that acts asIt is convenient to regard m not as a bilinear map toȦ from the Kronecker productȦ ×Ȧ, but as a linear map toȦ from the tensor productȦ ⊗ kȦ . In terms of the linear maps m and η, the axioms (1.1) of A readIt would actually be more precise to call the structure just described an algebra in the category Vect k of finite-dimensional k-vector spaces. When this formulation is adopted, the requirement of linearity of the maps η and m is already built in automatically, namely by merely demanding that they are allowed maps at all, i.e. that they are morphisms m ∈ Hom(Ȧ ⊗Ȧ,Ȧ) and η ∈ Hom(1,Ȧ) (1.4)

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“…A natural method for classifying objects in mathematics is via numerical invariants. In [NS1], Ng and Schauenburg introduced a class of invariants of spherical pivotal fusion categories (to be simply called spherical categories) called the higher Frobenius-Schur indicators. Let C denote a spherical category.…”

Section: Introductionmentioning

confidence: 99%

Indicators of Tambara–Yamagami categories and Gauss sums

Basak

Johnson

2015

Algebra Number Theory

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Abstract. We prove that the higher Frobenius-Schur indicators, introduced by Ng and Schauenburg, give a strong enough invariant to distinguish between any two TambaraYamagami fusion categories. Our proofs are based on computation of the higher indicators in terms of Gauss sums for certain quadratic forms on finite abelian groups and rely on the classification of quadratic forms on finite abelian groups, due to Wall.As a corollary to our work, we show that the state-sum invariants of a Tambara-Yamagami category determine the category as long as we restrict to Tambara-Yamagami categories coming from groups G whose order is not a power of 2. Turaev and Vainerman proved this result under the assumption that G has odd order and they conjectured that a similar result should hold for groups of even order. We also give an example to show that the assumption that |G| is not a power of 2, cannot be completely relaxed.

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Higher Frobenius-Schur indicators for pivotal categories

Cited by 83 publications

References 14 publications

The half-twist forU_q(g) representations

The half-twist forU_q(g) representations

The graphical calculus for ribbon categories: Algebras, modules, Nakayama automorphisms

Indicators of Tambara–Yamagami categories and Gauss sums

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Higher Frobenius-Schur indicators for pivotal categories

Cited by 83 publications

References 14 publications

The half-twist forUq(g) representations

The half-twist forUq(g) representations

The graphical calculus for ribbon categories: Algebras, modules, Nakayama automorphisms

Indicators of Tambara–Yamagami categories and Gauss sums

Contact Info

Product

Resources

About

The half-twist forU_q(g) representations

The half-twist forU_q(g) representations