Quantum Invariants of 3-Manifolds Associated With Classical Simple Lie Algebras

Turaev, Vladimir; Wenzl, Hans

doi:10.1142/s0129167x93000170

Cited by 101 publications

(111 citation statements)

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“…t ∈ T , and assume ε r (β t , β t ′ ) = ε r (β t ′ , β t ) * for all t ′ ∈ T , where, as usual, β t ′ denotes a representative endomorphism for each . So here Theorem 6.12 can be rephrased, roughly speaking, as "there is a non-degenerate braiding on the even vertices of the graphs D even ", which is a known result, see [33,12,40]. For SU (3) the corresponding statement is that there is a non-degenerate braiding associated to the triality zero vertices of Kostov's graphs, see [28, Fig.…”

Section: Non-degenerate Braidings On Orbifold Graphsmentioning

confidence: 97%

Modular Invariants, Graphs and α-Induction¶for Nets of Subfactors. III

Böckenhauer¹,

Evans²

1999

Commun. Math. Phys.

114

321

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In this paper we further develop the theory of α-induction for nets of subfactors, in particular in view of the system of sectors obtained by mixing the two kinds of induction arising from the two choices of braiding. We construct a relative braiding between the irreducible subsectors of the two "chiral" induced systems, providing a proper braiding on their intersection. We also express the principal and dual principal graphs of the local subfactors in terms of the induced sector systems. This extended theory is again applied to conformal or orbifold embeddings of SU (n) WZW models. A simple formula for the corresponding modular invariant matrix is established in terms of the two inductions, and we show that it holds if and only if the sets of irreducible subsectors of the two chiral induced systems intersect minimally on the set of marked vertices, i.e. on the "physical spectrum" of the embedding theory, or if and only if the canonical endomorphism sector of the conformal or orbifold inclusion subfactor is in the full induced system. We can prove either condition for all simple current extensions of SU (n) and many conformal inclusions, covering in particular all type I modular invariants of SU (2) and SU (3), and we conjecture that it holds also for any other conformal inclusion of SU (n) as well. As a by-product of our calculations, the dual principal graph for the conformal inclusion SU (3) 5 ⊂ SU (6) 1 is computed for the first time.

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Section: Non-degenerate Braidings On Orbifold Graphsmentioning

confidence: 97%

Modular Invariants, Graphs and α-Induction¶for Nets of Subfactors. III

Böckenhauer¹,

Evans²

1999

Commun. Math. Phys.

114

321

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“…In fact, specializations of the famous polynomial invariants of Jones [J], the six-authored paper [HOMFLY] and Kauffman [Kf] have been obtained in this way. Reshetikhin and Turaev [RT] used this connection to derive invariants of 3-manifolds from modular Hopf algebras, examples of which can be found among quantum groups at roots of unity (see [RT] and [TW1], much simplified by constructions in [A]). When Witten [Wi] introduced the notion of a topological quantum field theory (TQFT) relating ideas from quantum field theory to manifold invariants, non-trivial examples were immediately available from the constructions in [RT] (after reconciling notation).…”

Section: Introductionmentioning

confidence: 99%

From quantum groups to unitary modular tensor categories

Rowell¹

2006

Contemporary Mathematics

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Abstract. Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3-dimensional TQFTs. Although other constructions have since been found, quantum groups remain the most prolific source. Recently proposed applications to quantum computing have provided an impetus to understand and describe these examples as explicitly as possible, especially those that are "physically feasible." We survey the current status of the problem of producing unitary modular tensor categories from quantum groups, emphasizing explicit computations.

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“…Our method is based on the skein-theoretical construction of idempotents in the Birman-Murakami-Wenzl (BMW) algebras given in [2]. This work follows the program of Turaev and Wenzl [19,20]. We give four specifications of parameters α and s (entering the Kauffman skein relations) which lead to different series of modular categories.…”

Section: Introductionmentioning

confidence: 99%

Modular categories of types B, C and D

Beliakova¹,

Blanchet²

2001

Comment. Math. Helv.

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Abstract.We construct four series of modular categories from the two-variable Kauffman polynomial, without use of the representation theory of quantum groups at roots of unity. The specializations of this polynomial corresponding to quantum groups of types B, C and D produce series of pre-modular categories. One of them turns out to be modular and three others satisfy Bruguières' modularization criterion. For these four series we compute the Verlinde formulas, and discuss spin and cohomological refinements. Mathematics Subject Classification (2000). 57M25, 57R56.

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Quantum Invariants of 3-Manifolds Associated With Classical Simple Lie Algebras

Cited by 101 publications

References 0 publications

Modular Invariants, Graphs and α-Induction¶for Nets of Subfactors. III

Modular Invariants, Graphs and α-Induction¶for Nets of Subfactors. III

From quantum groups to unitary modular tensor categories

Modular categories of types B, C and D

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