Gauge invariants from the powers of antipodes

Negron, Cris; Ng, Siu-Hung

doi:10.2140/pjm.2017.291.439

Cited by 6 publications

(13 citation statements)

References 38 publications

(73 reference statements)

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“…8, 8.2, 8.3, and Theorem 7.4 below.) Our analyses of the Drinfeld element and antipode give positive answers to some general questions of [22] and [26] in the particular case of Belavin-Drinfeld twists of the small quantum group. We describe our result on irreducibles more explicitly below.…”

Section: Introductionmentioning

confidence: 86%

“…Implications for the antipode. In [22] the question was posed as to whether or not the order of the antipode and the traces of the powers of the antipode are preserved under twisting. The question was answered positively for Hopf algebras with the Chevalley property.…”

Section: The Drinfeld Element and Properties Of The Antipodementioning

confidence: 99%

“…We can also get invariance of the so-called regular object of [26,Sect. 5.4] using the condition of [22,Prop. 7.3 (ii)].…”

Section: The Drinfeld Element and Properties Of The Antipodementioning

confidence: 99%

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Small quantum groups associated to Belavin-Drinfeld triples

Negron

2018

Trans. Amer. Math. Soc.

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Abstract. For a simple Lie algebra g of type A, D, E we show that any Belavin-Drinfeld triple on the Dynkin diagram of g produces a collection of Drinfeld twists for Lusztig's small quantum group uq(g). These twists give rise to new finite-dimensional factorizable Hopf algebras, i.e. new small quantum groups. For any Hopf algebra constructed in this manner, we identify the group of grouplike elements, identify the Drinfeld element, and describe the irreducible representations of the dual in terms of the representation theory of the parabolic subalgebra(s) in g associated to the given Belavin-Drinfeld triple. We also produce Drinfeld twists of uq(g) which express a known algebraic group action on its category of representations, and pose a subsequent question regarding the classification of all twists.

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Section: Introductionmentioning

confidence: 86%

Section: The Drinfeld Element and Properties Of The Antipodementioning

confidence: 99%

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Small quantum groups associated to Belavin-Drinfeld triples

Negron

2018

Trans. Amer. Math. Soc.

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“…The F SZ properties for groups were introduced by Iovanov et al [4], and were inspired by invariants of representation categories of semisimple Hopf algebras [5,11,12]. These invariants and their generalizations have proven extremely useful in a wide range of areas; see [10] for a detailed discussion and references.…”

Section: Background and Notationmentioning

confidence: 99%

Some behaviors of FSZ groups under central products, central quotients, and regular wreath products

Keilberg¹

2019

Journal of Algebra

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We show that any group G with a non-F SZm quotient by a central cyclic subgroup also provides a non-F SZm group of order m|G| obtained as a central product of G with a cyclic group. We then construct, for every prime p > 3 and j ∈ N, an F SZ p j group F such that there is a central cyclic subgroup A with F/A not F SZ p j . We apply these results to regular wreath products to construct an F SZ p-group which is not F SZ + for any prime p > 3. These give the first known examples of F SZ groups that are not F SZ + . We are also able to prove a few partial results concerning the F SZ properties for the Sylow subgroups of symmetric groups. In the appendix we enumerate all non-F SZ groups of order 5 7 .

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“…The F SZ properties for groups, as introduced by Iovanov et al [4], arise from considerations of certain invariants of the representation categories of semisimple Hopf algebras known as higher Frobenius-Schur indicators [5,10,11]. See [9] for a detailed discussion of the many important uses and generalizations of these invariants. When applied to Drinfeld doubles of finite groups, these invariants are described entirely in group theoretical terms, and are in particular invariants of the group itself.…”

Section: Introductionmentioning

confidence: 99%

The FSZ properties of sporadic simple groups

Keilberg¹

2019

J. Algebra Appl.

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We investigate a possible connection between the F SZ properties of a group and its Sylow subgroups. We show that the simple groups G 2 (5) and S 6 (5), as well as all sporadic simple groups with order divisible by 5 6 are not F SZ, and that neither are their Sylow 5-subgroups. The groups G 2 (5) and HN were previously established as non-F SZ by Peter Schauenburg; we present alternative proofs. All other sporadic simple groups and their Sylow subgroups are shown to be F SZ. We conclude by considering all perfect groups available through GAP with order at most 10 6 , and show they are non-F SZ if and only if their Sylow 5-subgroups are non-F SZ.

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Gauge invariants from the powers of antipodes

Cited by 6 publications

References 38 publications

Small quantum groups associated to Belavin-Drinfeld triples

Small quantum groups associated to Belavin-Drinfeld triples

Some behaviors of FSZ groups under central products, central quotients, and regular wreath products

The FSZ properties of sporadic simple groups

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