Kauffman brackets, character varieties and triangulations of surfaces

Bonahon, Francis; Wong, Helen

doi:10.1090/conm/560/11099

Cited by 14 publications

(8 citation statements)

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“…By composition with the trace homomorphism Tr ω λ : S A (S) → Z ω λ provided by Theorem 1, one obtains a wide family of finite-dimensional representations of the skein algebra S A (S). These representations behave well with respect to the action of the mapping class group, and a great feature of the corresponding machinery is that it works even for closed surfaces [6,7,8]. In particular, the results of the current paper represent a key technical step in a long-term program to study the representation theory of the skein algebra S A (S); see [6] for a discussion.…”

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confidence: 80%

Quantum traces for representations of surface groups in SL₂(ℂ)

2011

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We relate two different quantizations of the character variety consisting of all representations of surface groups in SL 2 . One is the Kauffman skein algebra considered by Bullock, Frohman and Kania-Bartoszyńska, Przytycki and Sikora, and Turaev. The other is the quantum Teichmüller space introduced by Chekhov and Fock and by Kashaev. We construct a homomorphism from the skein algebra to the quantum Teichmüller space which, when restricted to the classical case, corresponds to the equivalence between these two algebras through trace functions. 14D20, 57M25, 57R56Let S be an oriented surface of finite topological type. The goal of this paper is to establish a connection between two quantizations of the character variety R SL 2 .C/ .S/ D fr W 1 .S/ ! SL 2 .C/g = = SL 2 .C/;consisting of all group homomorphisms r from the fundamental group 1 .S / to the Lie group SL 2 .C/, considered up to conjugation by elements of SL 2 .C/. The double bar indicates here that the quotient is taken in the algebraic geometric sense of geometric invariant theory.The first quantization was introduced by D Bullock, C Frohman, J Kania-Bartoszyńska [12], J Przytycki and A Sikora [32] and V Turaev [35] and uses the Kauffman skein algebra S A .S/. This algebra is obtained by considering the vector space freely generated by all isotopy classes of framed links in S OE0; 1, and then taking the quotient of this space under the Kauffman skein relation; see Section 3.1. What makes S A .S / a quantization of R SL 2 .C/ .S/ is that, when A D 1, the skein algebra S 1 .S/ has a natural identification with the commutative algebra of regular functions on R SL 2 .C/ .S/ and that, as A tends to 1, the lack of commutativity of S A .S / is infinitesimally measured by the Goldman-Weil-Petersson Poisson structure [19; 20; 31; 36] on R SL 2 .C/ .S /; see [35]. There is a similar situation when A D C1, in which case S C1 .S/ has a natural identification with the algebra of functions on a twisted version of R SL 2 .C/ .S/; see Section 3.2. Guo and Liu [21]. This quantization takes advantage of the fact that, if one restricts to matrices with real coefficients, a large subset of R SL 2 .R/ .S/ with nonempty interior has a natural identification with the Teichmüller (or Fricke-Klein) space T .S/, consisting of isotopy classes of all complete hyperbolic metrics on S . Starting with an ideal triangulation of the surface, Thurston [33;34] introduced for the Teichmüller space T .S/ a set of coordinates, called shear coordinates, in which the Goldman-Weil-Petersson form is expressed in a particularly simple way. The quantum Teichmüller space is a quantization of T .S/ that is based on these shear coordinates. As this construction requires the existence of an ideal triangulation, the surface must have at least one puncture.A natural conjecture is that these two quantizations are "essentially equivalent".In the classical cases where q D 1 and A D˙1, the correspondence is relatively clear because of the identifications of S˙1.S/ and T 1 S with algebras of functions...

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Quantum traces for representations of surface groups in SL₂(ℂ)

2011

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“…However, it is crucial for us to work in the context of principal coefficients, because our proofs in [MSW2] use the notion of g-vectors, which are defined in the case of principal coefficients. Matrix formulas have also appeared in related literature, including [ADSS,ARS,BW,BW2,Pen2,Pen3].…”

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confidence: 99%

Matrix Formulae and Skein Relations for Cluster Algebras from Surfaces

Musiker¹,

Williams²

2012

International Mathematics Research Notices

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Abstract. This paper concerns cluster algebras with principal coefficients A•(S, M ) associated to bordered surfaces (S, M ), and is a companion to a concurrent work of the authors with Schiffler [MSW2]. Given any (generalized) arc or loop in the surface -with or without self-intersections -we associate an element of (the fraction field of) A•(S, M ), using products of elements of P SL2(R). We give a direct proof that our matrix formulas for arcs and loops agree with the combinatorial formulas for arcs and loops in terms of matchings, which were given in [MSW,MSW2]. Finally, we use our matrix formulas to prove skein relations for the cluster algebra elements associated to arcs and loops. Our matrix formulas and skein relations generalize prior work of Fock and Goncharov [FG1,FG2,FG3], who worked in the coefficient-free case. The results of this paper will be used in [MSW2] in order to show that certain collections of arcs and loops comprise a vector-space basis for A•(S, M ).

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“…For every primitive 2Nroot of unity A, the Witten-Reshetikhin-Turaev representation ρ :This result can be compared with Roberts's proof [22] that, when N = 2p with p prime, the action of the mapping class group of S on the Witten-Reshetikhin-Turaev space V S is irreducible. In this special case, Theorem 1 can actually be deduced from some of the proofs of [22].Our interest in the irreducibility of the Witten-Reshetikhin-Turaev representation is motivated by [3, 4, 5, 6], where we initiated the systematic study of finite-dimensional irreducible representations of the skein algebra S A (S). In particular, when A is a 2N -root of unity with N odd, we associate to such an irreducible representation ρ : S A (S) → End(V ) an element r ρ of the character variety…”

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The Witten-Reshetikhin-Turaev representation of the Kauffman bracket skein algebra

Bonahon¹,

Wong²

2015

Proc. Amer. Math. Soc.

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For A a primitive 2N -root of unity with N odd, the Witten-Reshetikhin-Turaev topological quantum field theory provides a representation of the Kauffman skein algebra of a closed surface. We show that this representation is irreducible, and we compute its classical shadow in the sense of [4].The discovery of the Jones polynomial [9], and the simplification of its construction by Kauffman [10], were quickly followed by two originally unrelated developments. The first one was the introduction of the Kauffman skein module of an oriented 3-manifold by Turaev [23] and Przytycki [17]. A special case leads to the Kauffman skein algebra S A (S) of an oriented surface S which, when S is connected, was later interpreted as a quantization of the character variety R SL2(C) (S), consisting of the characters of all group homomorphisms π 1 (S) → SL 2 (C) [24,7, 8,18].Another development was Witten's interpretation [26] of the Jones polynomial within the framework of a topological quantum field theory. This topological quantum field theory point of view was formalized in mathematical terms by 20]. In particular, the Witten-Reshetikhin-Turaev topological quantum field theory leads, for every primitive 2N -root of unity A, to a representation ρ : S A (S) → End(V S ) of the skein algebra corresponding to this parameter A.The first result of this article is the following.Theorem 1. Let S be a connected closed oriented surface. For every primitive 2Nroot of unity A, the Witten-Reshetikhin-Turaev representation ρ :This result can be compared with Roberts's proof [22] that, when N = 2p with p prime, the action of the mapping class group of S on the Witten-Reshetikhin-Turaev space V S is irreducible. In this special case, Theorem 1 can actually be deduced from some of the proofs of [22].Our interest in the irreducibility of the Witten-Reshetikhin-Turaev representation is motivated by [3, 4, 5, 6], where we initiated the systematic study of finite-dimensional irreducible representations of the skein algebra S A (S). In particular, when A is a 2N -root of unity with N odd, we associate to such an irreducible representation ρ : S A (S) → End(V ) an element r ρ of the character variety

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Kauffman brackets, character varieties and triangulations of surfaces

Cited by 14 publications

References 32 publications

Quantum traces for representations of surface groups in SL₂(ℂ)

Quantum traces for representations of surface groups in SL₂(ℂ)

Matrix Formulae and Skein Relations for Cluster Algebras from Surfaces

The Witten-Reshetikhin-Turaev representation of the Kauffman bracket skein algebra

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Kauffman brackets, character varieties and triangulations of surfaces

Cited by 14 publications

References 32 publications

Quantum traces for representations of surface groups in SL2(ℂ)

Quantum traces for representations of surface groups in SL2(ℂ)

Matrix Formulae and Skein Relations for Cluster Algebras from Surfaces

The Witten-Reshetikhin-Turaev representation of the Kauffman bracket skein algebra

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Quantum traces for representations of surface groups in SL₂(ℂ)

Quantum traces for representations of surface groups in SL₂(ℂ)