Mappin class group actions on quantum doubles

Kerler, Thomas

doi:10.1007/bf02101554

Cited by 58 publications

(103 citation statements)

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“…Such a "deficient factorizability" suffices for constructing a modular group action on the g p + ,p − center Z. In accordance with [17,18,19], the action of S = ( 0 1 −1 0 ) ∈ SL(2, Z) on Z is constructed in terms of the Drinfeld and Radford maps χ and φ φ φ and is given by composing one of these with the inverse of the other,…”

Section: Theorem Multiplication In the Gmentioning

confidence: 99%

Kazhdan-Lusztig-dual quantum group for logarithimic extensions of Virasoro minimal models

Feigin

Gainutdinov

Semikhatov

et al. 2007

Journal of Mathematical Physics

181

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ABSTRACT. We derive and study a quantum group g p,q that is Kazhdan-Lusztig-dual to the W -algebra W p,q of the logarithmic (p, q) conformal field theory model. The algebra W p,q is generated by two currents W + (z) and W − (z) of dimension (2p−1)(2q−1), and the energy-momentum tensor T (z). The two currents generate a vertex-operator ideal R with the property that the quotient W p,q /R is the vertex-operator algebra of the (p, q) Virasoro minimal model. The number (2pq) of irreducible g p,q representations is the same as the number of irreducible W p,q -representations on which R acts nontrivially. We find the center of g p,q and show that the modular group representation on it is equivalent to the modular group representation on the W p,q characters and "pseudocharacters." The factorization of the g p,q ribbon element leads to a factorization of the modular group representation on the center. We also find the g p,q Grothendieck ring, which is presumably the "logarithmic" fusion of the (p, q) model. CONTENTS

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Section: Theorem Multiplication In the Gmentioning

confidence: 99%

Kazhdan-Lusztig-dual quantum group for logarithimic extensions of Virasoro minimal models

Feigin

Gainutdinov

Semikhatov

et al. 2007

Journal of Mathematical Physics

181

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“…Moreover, when the Hopf algebra is the finite dimensional quantum enveloping algebra at root of unity of some simple Lie group, T s contains at least one more element z RT which corresponds to the Reshetikhin-Turaev invariant. This fact was first observed by Hennings, and then, for the quantum sl (2), z RT was made explicit by Kerler in [8] (for completeness, in the appendix we present the derivation of z RT ). In an analogous way (though it won't be done here), one can see that Quinn's invariant can be derived in the HKR-framework from a triangular Hopf algebra over Z/p and a central element z Q = 1 in it.…”

Section: 1mentioning

confidence: 56%

“…The center of A is described in [8], where the following notations are used: (1) and F = c∈Z/p 1 c F (1) . Following [8] we define…”

Section: 4mentioning

confidence: 99%

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HKR-type invariants of 4–thickenings of 2–dimensional CW complexes

Bobtcheva

Messia

2003

Algebr. Geom. Topol.

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The HKR (Hennings-Kauffman-Radford) framework is used to construct invariants of 4-thickenings of 2-dimensional CW complexes under 2-deformations (1-and 2-handle slides and creations and cancellations of 1-2 handle pairs). The input of the invariant is a finite dimensional unimodular ribbon Hopf algebra A and an element in a quotient of its center, which determines a trace function on A. We study the subset T 4 of trace elements which define invariants of 4-thickenings under 2-deformations. In T 4 two subsets are identified : T 3 ⊂ T 4 , which produces invariants of 4-thickenings normalizable to invariants of the boundary, and T 2 ⊂ T 4 , which produces invariants of 4-thickenings depending only on the 2-dimensional spine and the second Whitney number of the 4-thickening. The case of the quantum sl(2) is studied in details. We conjecture that sl(2) leads to four HKR-type invariants and describe the corresponding trace elements. Moreover, the fusion algebra of the semisimple quotient of the category of representations of the quantum sl(2) is identified as a subalgebra of a quotient of its center.

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Bridged Links and Tangle Presentations of Cobordism Categories

Kerler

1999

Advances in Mathematics

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We develop a calculus of surgery data, called bridged links, which involves besides links also pairs of balls that describe one-handle attachments. As opposed to the usual link calculi of Kirby and others this description uses only elementary, local moves (namely modifications and isolated cancellations), and it is valid also on non-simply connected and disconnected manifolds. In particular, it allows us to give a presentation of a 3-manifold by doing surgery on any other 3-manifold with the same boundary. Bridged link presentations on unions of handlebodies are used to give a Cerf-theoretical derivation of presentations of 2+1-dimensional cobordisms categories in terms of planar ribbon tangles and their composition rules. As an application we give a different, more natural proof of the Matveev Polyak presentations of the mapping class groups, and, furthermore, find systematically surgery presentations of general mapping tori. We discuss a natural extension of the Reshetikhin Turaev invariant to the calculus of bridged links. Invariance follows now similar as for knot invariants from simple identifications of the elementary moves with elementary categorical relations for invariances or cointegrals, respectively. Hence, we avoid the lengthy computations and the unnatural Fenn Rourke reduction of the original proofs. Moreover, we are able to start from a much weaker``modularity''-condition, which implies the one of Turaev. Generalizations of the presentation to cobordisms of surfaces with boundaries are outlined. Academic PressContents

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Mappin class group actions on quantum doubles

Cited by 58 publications

References 10 publications

Kazhdan-Lusztig-dual quantum group for logarithimic extensions of Virasoro minimal models

Kazhdan-Lusztig-dual quantum group for logarithimic extensions of Virasoro minimal models

HKR-type invariants of 4–thickenings of 2–dimensional CW complexes

Bridged Links and Tangle Presentations of Cobordism Categories

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