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Representations of the quantum Teichmüller space, and invariants of surface diffeomorphisms
 GEOM. TOPOL
, 2004
"... We investigate the representation theory of the polynomial core T q S of the quantum Teichmüller space of a punctured surface S. This is a purely algebraic object, closely related to the combinatorics of the simplicial complex of ideal cell decompositions of S. Our main result is that irreducible ..."
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We investigate the representation theory of the polynomial core T q S of the quantum Teichmüller space of a punctured surface S. This is a purely algebraic object, closely related to the combinatorics of the simplicial complex of ideal cell decompositions of S. Our main result is that irreducible finitedimensional representations of T q S are classified, up to finitely many choices, by group homomorphisms from the fundamental group π1(S) to the isometry group of the hyperbolic 3–space H³. We exploit this connection between algebra and hyperbolic geometry to exhibit invariants of diffeomorphisms of S.
A uniqueness property for the quantization of Teichmüller spaces
, 2008
"... Chekhov, Fock and Kashaev introduced a quantization of the Teichmüller space T q (S) of a punctured surface S, and an exponential version of this construction was developed by Bonahon and Liu. The construction of the quantum Teichmüller space crucially depends on certain coordinate change isomorph ..."
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Cited by 12 (2 self)
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Chekhov, Fock and Kashaev introduced a quantization of the Teichmüller space T q (S) of a punctured surface S, and an exponential version of this construction was developed by Bonahon and Liu. The construction of the quantum Teichmüller space crucially depends on certain coordinate change isomorphisms between the ChekhovFock algebras associated to different ideal triangulations of S. We show that these coordinate change isomorphisms are essentially unique, once we require them to satisfy a certain number of natural conditions.
Quantum traces for representations of surface groups
 in SL2(C), Geom. Topol
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Quantum hyperbolic invariants of diffeomorphism of the punctured torus, in preparation
"... Abstract. An earlier article [1] introduced new invariants for pseudoAnosov diffeomorphisms of surface, based on the representation theory of the quantum Teichmüller space. We explicity compute these quantum hyperbolic invariants in the case of the 1–puncture torus and the 4–puncture sphere. 1. ..."
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Abstract. An earlier article [1] introduced new invariants for pseudoAnosov diffeomorphisms of surface, based on the representation theory of the quantum Teichmüller space. We explicity compute these quantum hyperbolic invariants in the case of the 1–puncture torus and the 4–puncture sphere. 1.
REPRESENTATIONS OF THE KAUFFMAN SKEIN ALGEBRA I: INVARIANTS AND MIRACULOUS CANCELLATIONS
, 2013
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Quantum Teichmüller spaces and Kashaev’s 6jsymbols
, 2007
"... The Kashaev invariants of 3–manifolds are based on 6j –symbols from the representation theory of the Weyl algebra, a Hopf algebra corresponding to the Borel subalgebra of Uq.sl.2; �//. In this paper, we show that Kashaev’s 6j –symbols are intertwining operators of local representations of quantum Te ..."
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The Kashaev invariants of 3–manifolds are based on 6j –symbols from the representation theory of the Weyl algebra, a Hopf algebra corresponding to the Borel subalgebra of Uq.sl.2; �//. In this paper, we show that Kashaev’s 6j –symbols are intertwining operators of local representations of quantum Teichmüller spaces. This relates Kashaev’s work with the theory of quantum Teichmüller space, which was developed by Chekhov–Fock, Kashaev and continued by Bonahon–Liu.
Factorization rules in quantum Teichmüller theory
"... Abstract. For a punctured surface S, a point of its Teichmüller space T(S) determines an irreducible representation of its quantization Tq(S). We analyze the behavior of these representations as one goes to infinity in T(S), or in the moduli space M(S) of the surface. The main result of this paper ..."
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Abstract. For a punctured surface S, a point of its Teichmüller space T(S) determines an irreducible representation of its quantization Tq(S). We analyze the behavior of these representations as one goes to infinity in T(S), or in the moduli space M(S) of the surface. The main result of this paper states that an irreducible representation of Tq(S) limits to a direct sum of representations of Tq(Sγ), where Sγ is obtained from S by pinching a multicurve γ to a set of nodes. The result is analogous to the factorization rule found in conformal field theory. Let S be an oriented surface of genus g obtained from a closed compact surface S by removing s punctures v1,..., vs. The Teichmüller space T(S) of S is the space of isotopy classes of complete hyperbolic metrics on S with finite area. It comes equipped with a natural Kähler metric, called the Weil–Petersson metric which is invariant under the action of the mapping class group MCG(S) onto T(S). A quantization of the Teichmüller space was successfully described by L. Chekhov and V. V. Fock in [8], and, in a
QUANTUM TEICHMÜLLER THEORY AND REPRESENTATIONS OF THE PURE BRAID GROUP
, 2008
"... Dedicated to the memory of XiaoSong Lin, an important mathematician and a great colleague Abstract. We adapt some of the methods of quantum Teichmüller theory to construct a family of representations of the pure braid group of the sphere. of a connected oriented surface S with negative Euler charac ..."
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Dedicated to the memory of XiaoSong Lin, an important mathematician and a great colleague Abstract. We adapt some of the methods of quantum Teichmüller theory to construct a family of representations of the pure braid group of the sphere. of a connected oriented surface S with negative Euler characteristic, following V. Fock, L. Chekhov [7, 8, 4] and R. Kashaev [12]. This is designed as a noncommutative deformation of the algebra of rational functions on the classical Teichmüller space of S, depending on a parameter q ∈ C ∗. It is a purely algebraic object, defined in terms of the combinatorics of the HarerPenner complex of the ideal cell decompositions of S. Finitedimensional representations of the quantum Teichmüller space exist only if q is a root of unity, and more precisely only if a power of −q is equal to −1. Whether one considers irreducible representations as in [3], or local representations as in [2], the main results have the same flavor. If −q is a primitive N–root of −1, a representation of the quantum Teichmüller space is classified up to isomorphism by: This article presents a variation of the constructions of [2, 3] on the finitedimensional representation theory of the quantum Teichmüller space. In these two earlier articles, the author and his collaborators considered the quantum Teichmüller space T q
QUANTUM TEICHMÜLLER SPACE AND KASHAEV ALGEBRA
, 2009
"... Kashaev algebra associated to a surface is a noncommutative deformation of the algebra of rational functions of Kashaev coordinates. For two arbitrary complex numbers, there is a generalized Kashaev algebra. The relationship between the shear coordinates and Kashaev coordinates induces a natural re ..."
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Kashaev algebra associated to a surface is a noncommutative deformation of the algebra of rational functions of Kashaev coordinates. For two arbitrary complex numbers, there is a generalized Kashaev algebra. The relationship between the shear coordinates and Kashaev coordinates induces a natural relationship between the quantum Teichmüller space and the generalized Kashaev algebra.