Quantum Groups at Roots of Unity and Modularity

Sawin, Stephen

doi:10.1142/s0218216506005160

Cited by 46 publications

(96 citation statements)

References 42 publications

(34 reference statements)

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“…VI of [17] 10 cf., e.g., [47,49]. Sometimes in the literature a different convention is used for the definition of Uq(g C ), which leads to the formula q := e πi D(k+cg ) where D is the quotient of the square lengths of the long and the short roots of g (cf., e.g., the second page of the introduction in [48])…”

Section: Algebraic Preliminaries 21 Concepts From Classical Lie Theorymentioning

confidence: 99%

Chern–simons Theory and the Quantum Racah Formula

Haro

Hahn

2013

Rev. Math. Phys.

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We generalize several results on Chern-Simons models on Σ × S 1 in the so-called "torus gauge" which were obtained in [33] (= arXiv:math-ph/0507040) to the case of general (simply-connected simple compact) structure groups and general link colorings. In particular, we give a non-perturbative evaluation of the Wilson loop observables corresponding to a special class of simple but non-trivial links and show that their values are given by Turaev's shadow invariant. As a byproduct we obtain a heuristic path integral derivation of the quantum Racah formula.

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Section: Algebraic Preliminaries 21 Concepts From Classical Lie Theorymentioning

confidence: 99%

Chern–simons Theory and the Quantum Racah Formula

Haro

Hahn

2013

Rev. Math. Phys.

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“…Many algebraic examples of modular categories, see, for example, [16], have k = Q(ε), a cyclotomic extension of the rationals, far from algebraically closed, and nevertheless End(V j ) ∼ = k for all simple objects. …”

Section: Modular Categoriesmentioning

confidence: 99%

Tannaka–Kreıˇn reconstruction and a characterization of modular tensor categories

Pfeiffer

2009

Journal of Algebra

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We show that every modular category is equivalent as an additive ribbon category to the category of finite-dimensional comodules of a Weak Hopf Algebra. This Weak Hopf Algebra is finitedimensional, split cosemisimple, weakly cofactorizable, coribbon and has trivially intersecting base algebras. In order to arrive at this characterization of modular categories, we develop a generalization of Tannaka-Kreǐn reconstruction to the long version of the canonical forgetful functor which is lax and oplax monoidal, but not in general strong monoidal, thereby avoiding all the difficulties related to non-integral Frobenius-Perron dimensions. In the more general case of a finitely semisimple additive ribbon category, not necessarily modular, the reconstructed Weak Hopf Algebra is finite-dimensional, split cosemisimple, coribbon and has trivially intersecting base algebras.

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“…In fact, much more is true (see Andersen [9], Andersen and Paradowski [10], Chari and Pressley [40], and Sawin [136]):…”

Section: Corollary 1731mentioning

confidence: 99%

Introduction to the Gopakumar–Vafa Large N Duality

Auckly¹,

Koshkin²

2007

Geometry and Topology Monographs

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Gopakumar-Vafa Large N Duality is a correspondence between Chern-Simons invariants of a link in a 3-manifold and relative Gromov-Witten invariants of a 6-dimensional symplectic manifold relative to a Lagrangian submanifold. We address the correspondence between the Chern-Simons free energy of S 3 with no link and the Gromov-Witten invariant of the resolved conifold in great detail. This case avoids mathematical difficulties in formulating a definition of relative Gromov-Witten invariants, but includes all of the important ideas.There is a vast amount of background material related to this duality. We make a point of collecting all of the background material required to check this duality in the case of the 3-sphere, and we have tried to present the material in a way complementary to the existing literature. This paper contains a large section on Gromov-Witten theory and a large section on quantum invariants of 3-manifolds. It also includes some physical motivation, but for the most part it avoids physical terminology.

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Quantum Groups at Roots of Unity and Modularity

Cited by 46 publications

References 42 publications

Chern–simons Theory and the Quantum Racah Formula

Chern–simons Theory and the Quantum Racah Formula

Tannaka–Kreıˇn reconstruction and a characterization of modular tensor categories

Introduction to the Gopakumar–Vafa Large N Duality

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