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18
On Associators and the GrothendieckTeichmüller Group I
, 1998
"... . We present a formalism within which the relationship (discovered by Drinfel'd in [Dr1, Dr2]) between associators (for quasitriangular quasiHopf algebras) and (a variant of) the GrothendieckTeichmuller group becomes simple and natural, leading to a simplication of Drinfel'd's original work. In p ..."
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Cited by 25 (4 self)
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. We present a formalism within which the relationship (discovered by Drinfel'd in [Dr1, Dr2]) between associators (for quasitriangular quasiHopf algebras) and (a variant of) the GrothendieckTeichmuller group becomes simple and natural, leading to a simplication of Drinfel'd's original work. In particular, we reprove that rational associators exist and can be constructed iteratively, though the proof itself still depends on the apriori knowledge that a notnecessarilyrational associator exists. Contents 1. Introduction 1 1.1. Reminders about quasitriangular quasiHopf algebras 1 1.2. What we do 2 1.3. Acknowledgement 4 2. The basic denitions 4 2.1. Parenthesized braids and GT 4 2.2. Parenthesized chord diagrams and GRT 8 3. Isomorphisms and associators 11 4. The Main Theorem 15 4.1. The statement, consequences, and rst reduction 15 4.2. More on the group \ GRT 15 4.3. The second reduction 18 4.4. A cohomological interlude 19 4.5. Proof of the semiclassical hexagon equation 20...
The Alexander polynomial and finite type 3manifold invariants
 Math. Ann
"... Abstract. Using elementary counting methods, we calculate the universal perturbative invariant (also known as the LMO invariant) of a 3manifold M, satisfying H1(M, Z) = Z, in terms of the Alexander polynomial of M. We show that +1 surgery on a knot in the 3sphere induces an injective map from fin ..."
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Cited by 19 (7 self)
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Abstract. Using elementary counting methods, we calculate the universal perturbative invariant (also known as the LMO invariant) of a 3manifold M, satisfying H1(M, Z) = Z, in terms of the Alexander polynomial of M. We show that +1 surgery on a knot in the 3sphere induces an injective map from finite type invariants of integral homology 3spheres to finite type invariants of knots. We also show that weight systems of degree 2m on knots, obtained by applying finite type 3m invariants of integral homology 3spheres, lie in the algebra of AlexanderConway weight systems, thus answering the questions raised in [Ga].
The Kontsevich Integral And Milnor's Invariants
, 1998
"... . A formula for computing the Milnor (concordance) invariants from the Kontsevich integral is obtained. The reduced Kontsevich integral (with values in the quotient by all loop diagrams) is shown to be the universal concordance invariant of finite type. Some applications are discussed. Contents Int ..."
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Cited by 10 (4 self)
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. A formula for computing the Milnor (concordance) invariants from the Kontsevich integral is obtained. The reduced Kontsevich integral (with values in the quotient by all loop diagrams) is shown to be the universal concordance invariant of finite type. Some applications are discussed. Contents Introduction 1 1. Review of basic concepts and notation. 3 2. The vector space A(X). 6 3. The universal invariant Z(T ). 7 4. The Hopfian monoidal category quotients A h and A t . 10 5. Review of Milnor's ¯ invariants. 11 6. The reduced Kontsevich integral Z t and Milnor's invariants. 12 7. A special case. 14 8. The first nonvanishing Milnor invariants and the lattice K n (l). 15 9. Vanishing results. 17 10. The case of homotopy Milnor invariants. 18 11. Proof of Theorem 6.1. 20 12. A global formula for Milnor's invariants. 22 13. Realizing primitive diagrams. 24 14. The concordance invariance of Z t for tangles. 26 15. The universal finite type concordance invariant. 28 16. Appendix o...
The Kontsevich integral and algebraic structures on the space of diagrams, from: “Knots in Hellas ’98”, Series on Knots and Everything 24, World Scientific (2000) 530–546
 Department of Pure Mathematics, University of Sheffield
, 2002
"... This paper is part expository and part presentation of calculational results. The target space of the Kontsevich integral for knots is a space of diagrams; this space has various algebraic structures which are described here. These are utilized with Le’s theorem on the behaviour of the Kontsevich in ..."
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Cited by 8 (2 self)
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This paper is part expository and part presentation of calculational results. The target space of the Kontsevich integral for knots is a space of diagrams; this space has various algebraic structures which are described here. These are utilized with Le’s theorem on the behaviour of the Kontsevich integral under cabling and with the MelvinMorton Theorem, to obtain, in the Kontsevich integral for torus knots, both an explicit expression up to degree five and the general coefficients of the wheel diagrams.
Biquantization of Lie bialgebras
 University of California, Berkeley
, 1990
"... Abstract. For any finitedimensional Lie bialgebra g, we construct a bialgebra Au,v(g) over the ring C[u][[v]], which quantizes simultaneously the universal enveloping bialgebra U(g), the bialgebra dual to U(g ∗), and the symmetric bialgebra S(g). Following [Tur89], we call Au,v(g) a biquantization ..."
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Cited by 7 (0 self)
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Abstract. For any finitedimensional Lie bialgebra g, we construct a bialgebra Au,v(g) over the ring C[u][[v]], which quantizes simultaneously the universal enveloping bialgebra U(g), the bialgebra dual to U(g ∗), and the symmetric bialgebra S(g). Following [Tur89], we call Au,v(g) a biquantization of S(g). We show that the bialgebra Au,v(g ∗ ) quantizing U(g ∗), U(g) ∗ , and S(g ∗ ) is essentially dual to the bialgebra obtained from Au,v(g) by exchanging u and v. Thus, Au,v(g) contains all information about the quantization of g. Our construction extends Etingof and Kazhdan’s onevariable quantization of U(g) [EK96].
Bottom tangles and universal invariants
, 2006
"... A bottom tangle is a tangle in a cube consisting only of arc components, each of which has the two endpoints on the bottom line of the cube, placed next to each other. We introduce a subcategory B of the category of framed, oriented tangles, which acts on the set of bottom tangles. We give a finite ..."
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Cited by 5 (2 self)
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A bottom tangle is a tangle in a cube consisting only of arc components, each of which has the two endpoints on the bottom line of the cube, placed next to each other. We introduce a subcategory B of the category of framed, oriented tangles, which acts on the set of bottom tangles. We give a finite set of generators of B, which provides an especially convenient way to generate all the bottom tangles, and hence all the framed, oriented links, via closure. We also define a kind of “braided Hopf algebra action ” on the set of bottom tangles. Using the universal invariant of bottom tangles associated to each ribbon Hopf algebra H, we define a braided functor J from B to the category ModH of left H–modules. The functor J, together with the set of generators of B, provides an algebraic method to study the range of quantum invariants of links. The braided Hopf algebra action on bottom tangles is mapped by J to the standard braided Hopf algebra structure for H in ModH. Several notions in knot theory, such as genus, unknotting number, ribbon knots, boundary links, local moves, etc are given algebraic interpretations in the setting involving the category B. The functor J provides a convenient way to study the relationships between these notions and quantum invariants.
Invariants de Vassiliev pour les entrelacs dans S³ et dans les variétés de dimension trois
, 1998
"... Algebra, Pergamon (1970), 329358. [Dri] V. G. Drinfeld, On quasitriangular quasiHopf algebras and a group closely connected with Gal(Q=Q), Algebra i Analiz 2:4 (1990), 149181. English transl.: Leningrad Math. J. 2 (1991), 829860. [Kas] C. Kassel, Quantum groups, GTM 155, SpringerVerlag, New ..."
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Cited by 4 (1 self)
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Algebra, Pergamon (1970), 329358. [Dri] V. G. Drinfeld, On quasitriangular quasiHopf algebras and a group closely connected with Gal(Q=Q), Algebra i Analiz 2:4 (1990), 149181. English transl.: Leningrad Math. J. 2 (1991), 829860. [Kas] C. Kassel, Quantum groups, GTM 155, SpringerVerlag, New York 1995. [KaT] C. Kassel and V. Turaev, Chord diagram invariants of tangles and graphs, Duke Math. J. 92, no. 3 (1998), 497552. [LM1] T. Q. T. Le and J. Murakami, Kontsevich integral for Kauffman polynomial, preprint MaxPlanckInstitut Bonn, 1993. [LM2] T. Q. T. Le and J. Murakami, The universal VassilievKontsevich invariant for framed oriented links, Comp. Math. 102 (1996), 4164. [LM3] T. Q. T. Le and J. Murakami, Parallel version of the universal Vassiliev Kontsevich invariant, J. Pure and Appl. Algebra 121 (1997), 271291. [LMO] T. Q. T. Le, J. Murakami and T. Ohtsuki, On a universal perturbative invariant of 3manifolds, Topology 373 (1998), 539574. [Prz] J. H. Przytyck...
The LMOinvariant of 3manifolds of rank one and the Alexander polynomial, preprint math.QA/0002040
"... Abstract. We prove that the LMOinvariant of a 3manifold of rank one is determined by the Alexander polynomial of the manifold, and conversely, that the Alexander polynomial is determined by the LMOinvariant. Furthermore, we show that the Alexander polynomial of a nullhomologous knot in a rationa ..."
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Cited by 4 (0 self)
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Abstract. We prove that the LMOinvariant of a 3manifold of rank one is determined by the Alexander polynomial of the manifold, and conversely, that the Alexander polynomial is determined by the LMOinvariant. Furthermore, we show that the Alexander polynomial of a nullhomologous knot in a rational homology 3sphere can be obtained by composing the weight system of the Alexander polynomial with the ˚Arhus invariant of knots.
Almost integral TQFTs from simple Lie algebras. Submitted for publication. Corrado De Concini and Victor G. Kac. Representations of quantum groups at roots of 1
 In Operator algebras, unitary representations, enveloping algebras, and invariant theory
, 1989
"... Abstract. Almost integral TQFT was introduced by Gilmer [G]. For each simple Lie algebra g and some prime integer we associate an almost integral TQFT which derives the projective WittenReshetikhinTuraev invariant τ Pg for closed 3manifolds. As a corollary, one can show that τ Pg is an algebraic ..."
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Cited by 3 (2 self)
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Abstract. Almost integral TQFT was introduced by Gilmer [G]. For each simple Lie algebra g and some prime integer we associate an almost integral TQFT which derives the projective WittenReshetikhinTuraev invariant τ Pg for closed 3manifolds. As a corollary, one can show that τ Pg is an algebraic integer for certain prime integers. The result in satisfies some Murasugi type equivalence relation if M is a homology sphere and admits a cyclic group action with fixed point set a circle. this paper can be used to prove that τ Pg M 1.
PACIFIC JOURNAL OF MATHEMATICS Vol. 195, No. 2, 2000 BIQUANTIZATION OF LIE BIALGEBRAS
"... For any finitedimensional Lie bialgebra g,we construct a bialgebra Au,v(g) over the ring C[u][[v]],which quantizes simultaneously the universal enveloping bialgebra U(g),the bialgebra dual to U(g ∗),and the symmetric bialgebra S(g). Following Turaev,we call Au,v(g) a biquantization of S(g). We show ..."
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For any finitedimensional Lie bialgebra g,we construct a bialgebra Au,v(g) over the ring C[u][[v]],which quantizes simultaneously the universal enveloping bialgebra U(g),the bialgebra dual to U(g ∗),and the symmetric bialgebra S(g). Following Turaev,we call Au,v(g) a biquantization of S(g). We show that the bialgebra Au,v(g ∗ ) quantizing U(g ∗), U(g) ∗,and S(g ∗ ) is essentially dual to the bialgebra obtained from Au,v(g) by exchanging u and v. Thus, Au,v(g) contains all information about the quantization of g. Our construction extends Etingof and Kazhdan’s onevariable quantization of U(g). Résumé. Etant donné une bigèbre de Lie g de dimension finie, nous construisons une C[u][[v]]bigèbre Au,v(g) qui quantifie simultanément la bigèbre enveloppante U(g), la bigèbre duale de U(g ∗ ) et la bigèbre symétrique S(g). Suivant Turaev, nous appelons Au,v(g) une biquantification de S(g). Nous montrons que la bigèbre Au,v(g ∗ ) qui quantifie U(g ∗), U(g) ∗ et S(g ∗ ) est en