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759
Quantization of Lie bialgebras
, 1996
"... This paper is a continuation of [EK14]. The goal of this paper is to define and study the notion of a quantum vertex operator algebra (VOA) in the setting of the formal deformation theory and give interesting examples of such algebras. Our definition of a quantum VOA is based on the ideas of the pa ..."
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Cited by 111 (15 self)
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This paper is a continuation of [EK14]. The goal of this paper is to define and study the notion of a quantum vertex operator algebra (VOA) in the setting of the formal deformation theory and give interesting examples of such algebras. Our definition of a quantum VOA is based on the ideas of the paper [FrR]. The first chapter of our paper is devoted to the general theory of quantum VOAs. For simplicity we consider only bosonic algebras, but all the definitions and results admit a straightforward generalization to the supercase. We start with the version of the definition of a VOA in which the main axiom is the locality (commutativity) axiom. To obtain a quantum deformation of this definition, we replace the locality axiom with the Slocality axiom, where S is a shiftinvariant unitary solution of the quantum YangBaxter equation (the other axioms are unchanged). We call the obtained structure a braided VOA. However, a braided VOA does not necessarily satisfy the associativity property, which is one of the main properties of a usual VOA. More precisely, instead of associativity it satisfies a quasiassociativity identity, which differs from associativity
THE COLORED JONES POLYNOMIALS AND THE SIMPLICIAL VOLUME OF A Knot
, 1999
"... We show that the set of colored Jones polynomials and the set of generalized Alexander polynomials defined by Akutsu, Deguchi and Ohtsuki intersect nontrivially. Moreover it is shown that the intersection is (at least includes) the set of Kashaev’s quantum dilogarithm invariants for links. Theref ..."
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Cited by 101 (10 self)
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We show that the set of colored Jones polynomials and the set of generalized Alexander polynomials defined by Akutsu, Deguchi and Ohtsuki intersect nontrivially. Moreover it is shown that the intersection is (at least includes) the set of Kashaev’s quantum dilogarithm invariants for links. Therefore Kashaev’s conjecture can be restated as follows: The colored Jones polynomials determine the hyperbolic volume for a hyperbolic knot. Modifying this, we propose a stronger conjecture: The colored Jones polynomials determine the simplicial volume for any knot. If our conjecture is true, then we can prove that a knot is trivial if and only if all of its Vassiliev invariants are trivial.
Faulttolerant quantum computation by anyons
, 2003
"... A twodimensional quantum system with anyonic excitations can be considered as a quantum computer. Unitary transformations can be performed by moving the excitations around each other. Measurements can be performed by joining excitations in pairs and observing the result of fusion. Such computation ..."
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Cited by 94 (3 self)
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A twodimensional quantum system with anyonic excitations can be considered as a quantum computer. Unitary transformations can be performed by moving the excitations around each other. Measurements can be performed by joining excitations in pairs and observing the result of fusion. Such computation is faulttolerant by its physical nature.
Algebras and Hopf algebras IN BRAIDED CATEGORIES
, 1995
"... This is an introduction for algebraists to the theory of algebras and Hopf algebras in braided categories. Such objects generalise superalgebras and superHopf algebras, as well as colourLie algebras. Basic facts about braided categories C are recalled, the modules and comodules of Hopf algebras i ..."
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Cited by 87 (13 self)
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This is an introduction for algebraists to the theory of algebras and Hopf algebras in braided categories. Such objects generalise superalgebras and superHopf algebras, as well as colourLie algebras. Basic facts about braided categories C are recalled, the modules and comodules of Hopf algebras in such categories are studied, the notion of ‘braidedcommutative ’ or ‘braidedcocommutative ’ Hopf algebras (braided groups) is reviewed and a fully diagrammatic proof of the reconstruction theorem for a braided group Aut (C) is given. The theory has important implications for the theory of quasitriangular Hopf algebras (quantum groups). It also includes important examples such as the degenerate Sklyanin algebra and the quantum plane.
Elliptic quantum groups
 In Proc. XIth International Congress of Mathematical Physics
, 1995
"... This note gives an account of a construction of an “elliptic quantum group” associated with each simple classical Lie algebra. It is closely related to elliptic face models of statistical mechanics, and, in its semiclassical limit, to the WessZuminoWitten model of conformal field theory on tori. M ..."
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Cited by 83 (9 self)
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This note gives an account of a construction of an “elliptic quantum group” associated with each simple classical Lie algebra. It is closely related to elliptic face models of statistical mechanics, and, in its semiclassical limit, to the WessZuminoWitten model of conformal field theory on tori. More details are presented in [Fe] and complete proofs will appear in a separate publication. Quantum groups (DrinfeldJimbo quantum enveloping algebras, Yangians, Sklyanin algebras, see [D], [Sk]) are the algebraic structures underlying integrable models of statistical mechanics and 2dimensional conformal field theory, and found applications in several other contexts. However, from the point of view of statistical mechanics, the picture is not quite complete. In particular, elliptic interactionroundaface models of statistical mechanics have sofar escaped a description in terms of quantum groups (expect in the slN case). In this paper, we give such a description. It is hoped that the construction will shed light in other contexts, such as a description of the category of representation of quantum affine Kac–Moody algebras, or the elliptic version of Macdonald’s theory. Our definition is motivated by the following known construction that links conformal field theory to the semiclassical version of quantum groups. Conformal blocks of WZW conformal field theory on the plane obey the consistent system of KnizhnikZamolodchikov (KZ) differential equations for a function u(z1,...,zn) taking values in the tensor product of n finite dimensional representations of a simple Lie algebra g [KZ]: ∂ziu = ∑ r(zi − zj) (ij) u (1) j:j̸=i Here, the “classical rmatrix ” r(z) is the tensor C/z, where C ∈ g ⊗ g is a symmetric invariant tensor. We use the notation X (i) , for X ∈ g or End(Vi), to
BEYOND SUPERSYMMETRY AND QUANTUM SYMMETRY (AN INTRODUCTION TO BRAIDEDGROUPS AND BRAIDEDMATRICES)
, 1993
"... ..."
Yangians And Classical Lie Algebras
"... Introduction The term `Yangian' was introduced by V. G. Drinfeld to specify quantum groups related to rational solutions of the classical YangBaxter equation; see Belavin Drinfeld [BD1,BD2] for the description of these solutions. In Drinfeld [D1] for each simple finitedimensional Lie algebra a ..."
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Cited by 71 (16 self)
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Introduction The term `Yangian' was introduced by V. G. Drinfeld to specify quantum groups related to rational solutions of the classical YangBaxter equation; see Belavin Drinfeld [BD1,BD2] for the description of these solutions. In Drinfeld [D1] for each simple finitedimensional Lie algebra a, a certain Hopf algebra Y(a) was constructed so that Y(a) is a deformation of the universal enveloping algebra for the polynomial current Lie algebra a[x]. An alternative description of the algebra Y(a) was given in Drinfeld [D3]; see Theorem 1 therein. Prior to the intruduction of the Hopf algebra Y(a) in Drinfeld [D1], the algebra which may be called the Yangian for the reductive Lie algebra gl(N) and may be denoted by Y , was considered in the works of mathematical physicists from St.Petersburg; see for instance TakhtajanFaddeev [TF]. The latter algebra is a deformation of the universal enveloping algebra U . Representations of were studied in KulishReshetikhinSklya
Quantum symmetry groups of finite spaces
 Comm. Math. Phys
, 1998
"... on the occasion of his sixtieth birthday Abstract. We determine the quantum automorphism groups of finite spaces. These are compact matrix quantum groups in the sense of Woronowicz. ..."
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Cited by 70 (4 self)
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on the occasion of his sixtieth birthday Abstract. We determine the quantum automorphism groups of finite spaces. These are compact matrix quantum groups in the sense of Woronowicz.
QuasiHopf twistors for elliptic quantum groups
"... Dedicated to Professor Mikio Sato on the occasion of his seventieth birthday The YangBaxter equation admits two classes of elliptic solutions, the vertex type and the face type. On the basis of these solutions, two types of elliptic quantum groups have been introduced (Foda et al.[1], Felder [2]). ..."
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Cited by 66 (12 self)
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Dedicated to Professor Mikio Sato on the occasion of his seventieth birthday The YangBaxter equation admits two classes of elliptic solutions, the vertex type and the face type. On the basis of these solutions, two types of elliptic quantum groups have been introduced (Foda et al.[1], Felder [2]). Frønsdal [3, 4] made a penetrating observation that both of them are quasiHopf algebras, obtained by twisting the standard quantum affine algebra Uq(g). In this paper we present an explicit formula for the twistors in the form of an infinite product of the universal R matrix of Uq(g). We also prove the shifted cocycle condition for the twistors, thereby completing Frønsdal’s findings. This construction entails that, for generic values of the deformation parameters, representation theory for Uq(g) carries over to the elliptic algebras, including such objects as evaluation modules, highest weight modules and vertex operators. In particular, we confirm the conjectures of Foda et al. concerning the elliptic algebra Aq,p(̂sl2). qalg/9712029 (to appear in Transformation Groups)