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22
Finite type invariants of Wknotted Objects: From Alexander to Kashiwara and Vergne
, 2008
"... Wknots, and more generally, wknotted objects (wbraids, wtangles, etc.) make a class of knotted objects which is wider but weaker than their “ordinary” counterparts. To get (say) wknots from ordinary knots, one has to allow nonplanar “virtual” knot diagrams, hence enlarging the the base set of ..."
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Wknots, and more generally, wknotted objects (wbraids, wtangles, etc.) make a class of knotted objects which is wider but weaker than their “ordinary” counterparts. To get (say) wknots from ordinary knots, one has to allow nonplanar “virtual” knot diagrams, hence enlarging the the base set of knots. But then one imposes a new relation, the “overcrossings commute ” relation, further beyond the ordinary collection of Reidemeister moves, making wknotted objects a bit weaker once again. The group of wbraids was studied (under the name “welded braids”) by Fenn, Rimanyi and Rourke [FRR] and was shown to be isomorphic to the McCool group [Mc] of “basisconjugating” automorphisms of a free group Fn — the smallest subgroup of Aut(Fn) that contains both braids and permutations. Brendle and Hatcher [BH], in work that traces back to Goldsmith [Gol], have shown this group to be a group of movies of flying rings in R 3. Satoh [Sa] studied several classes of wknotted objects (under the name “weaklyvirtual”) and has shown them to be closely related to certain classes of knotted surfaces in R 4. So wknotted objects are topologically and algebraically interesting. In this article we study finite type invariants of several classes of wknotted objects.
QUASICOXETER ALGEBRAS, DYNKIN DIAGRAM COHOMOLOGY AND QUANTUM WEYL GROUPS
, 2005
"... ... independently De Concini (unpublished), conjectured that the monodromy of the Casimir connection ∇C introduced in [MTL] is described by Lusztig’s quantum Weyl group operators. This conjecture was proved in [TL1] for all representations of the Lie algebra g = sln and in [TL2] for a number of pai ..."
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... independently De Concini (unpublished), conjectured that the monodromy of the Casimir connection ∇C introduced in [MTL] is described by Lusztig’s quantum Weyl group operators. This conjecture was proved in [TL1] for all representations of the Lie algebra g = sln and in [TL2] for a number of pairs (g, V) including vector and spin representations of classical Lie algebras and the adjoint representation of all complex, simple Lie algebras. The aim of this paper, and of its sequel [TL4] is to prove this conjecture for all g. Our strategy is inspired by Drinfeld’s proof of the equivalence of the monodromy of the Knizhnik–Zamolodchikov equations for g and the R–matrix representations coming from the quantum group U�g. It relies on the use of quasi–Coxeter algebras, which are to the generalised braid group of type g what Drinfeld’s quasitriangular quasibialgebras are to Artin’s braid groups Bn. Using this notion, and the associated deformation cohomology, which we call Dynkin diagram
A cohomological construction of quantization functors of Lie bialgebras, math.QA/0212325
"... Abstract. We propose a variant to the EtingofKazhdan construction of quantization functors. We construct the twistor JΦ associated to an associator Φ using cohomological techniques. We then introduce a criterion ensuring that the “left Hopf algebra ” of a quasitriangular QUE algebra is flat. We pro ..."
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Cited by 8 (6 self)
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Abstract. We propose a variant to the EtingofKazhdan construction of quantization functors. We construct the twistor JΦ associated to an associator Φ using cohomological techniques. We then introduce a criterion ensuring that the “left Hopf algebra ” of a quasitriangular QUE algebra is flat. We prove that this criterion is satisfied at the universal level. This provides a construction of quantization functors, equivalent to the EtingofKazhdan construction. Introduction. In [9, 10], Etingof and Kazhdan constructed quantization functors of Lie bialgebras over a field K of characteristic 0. Such a functor is a morphism of props Q: Bialg → ̂S • (̂LBA) with suitable classical limit properties. Here Bialg is the prop of bialgebras and ̂LBA is a completion of the prop of Lie bialgebras. Each quantization functor gives rise to a functor LBA → QUE from the category of Lie bialgebras to that of quantized universal enveloping algebra,
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...
COMPRESSED DRINFELD ASSOCIATORS
, 2004
"... Drinfeld associator is a key tool in computing the Kontsevich integral of knots. A Drinfeld associator is a series in two noncommuting variables, satisfying highly complicated algebraic equations — hexagon and pentagon. The logarithm of a Drinfeld associator lives in the Lie algebra L generated by ..."
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Cited by 4 (2 self)
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Drinfeld associator is a key tool in computing the Kontsevich integral of knots. A Drinfeld associator is a series in two noncommuting variables, satisfying highly complicated algebraic equations — hexagon and pentagon. The logarithm of a Drinfeld associator lives in the Lie algebra L generated by the symbols a, b, c modulo [a, b] = [b, c] = [c, a]. The main result is a description of compressed associators that satisfy the compressed pentagon and hexagon in the quotient L / [ [L, L], [L, L] ]. The key ingredient is an explicit form of CampbellBakerHausdorff formula in the case when all commutators commute.
The Drinfeld associator of gl(11
"... We determine explicitly a rational even Drinfeld associator Φ in a completion of the universal enveloping algebra of the Lie superalgebra gl(11) ⊕3. More generally, we define a new algebra of trivalent diagrams that has a unique even horizontal grouplike Drinfeld associator Φ. The associator Φ is ..."
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Cited by 3 (0 self)
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We determine explicitly a rational even Drinfeld associator Φ in a completion of the universal enveloping algebra of the Lie superalgebra gl(11) ⊕3. More generally, we define a new algebra of trivalent diagrams that has a unique even horizontal grouplike Drinfeld associator Φ. The associator Φ is mapped to Φ by a weight system. As a related result of independent interest, we show how O. Viro’s generalization ∆ 1 of the multivariable Alexander polynomial can be obtained from the universal Vassiliev invariant of trivalent graphs. We determine Φ by using the invariant ∆ 1 ( ) of a planar tetrahedron.
Conjectured enumeration of Vassiliev invariants
, 1997
"... Abstract A rational Ansatz is proposed for the generating function ∑ j,k β2j+k,2jxjy k, where βm,u is the number of primitive chinese character diagrams with u univalent and 2m − u trivalent vertices. For Pm: = ∑ u≥2 βm,u, the conjecture leads to the sequence ..."
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Cited by 3 (0 self)
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Abstract A rational Ansatz is proposed for the generating function ∑ j,k β2j+k,2jxjy k, where βm,u is the number of primitive chinese character diagrams with u univalent and 2m − u trivalent vertices. For Pm: = ∑ u≥2 βm,u, the conjecture leads to the sequence
Quantization of coboundary Lie bialgebras
 Annals of Math., math.QA/0603740
"... Abstract. We show that any coboundary Lie bialgebra can be quantized. For this, we prove that: (a) EtingofKazhdan quantization functors are compatible with Lie bialgebra twists, and (b) if such a quantization functor corresponds to an even associator, then it is also compatible with the operation o ..."
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Cited by 3 (2 self)
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Abstract. We show that any coboundary Lie bialgebra can be quantized. For this, we prove that: (a) EtingofKazhdan quantization functors are compatible with Lie bialgebra twists, and (b) if such a quantization functor corresponds to an even associator, then it is also compatible with the operation of taking coopposites. We also use the relation between the EtingofKazhdan construction of quantization functors and the alternative approach to this problem, which was established in a previous work. Let k be a field of characteristic 0. Unless specified otherwise, “algebra”, “vector space”, etc., means “algebra over k”, etc.
HOMOMORPHIC EXPANSIONS FOR KNOTTED TRIVALENT GRAPHS
"... Abstract. It had been known since old times [MO, CL, Da] that there exists a universal finite type invariant (“an expansion”) Z old for Knotted Trivalent Graphs (KTGs), and that it can be chosen to intertwine between some of the standard operations on KTGs and their chorddiagrammaticcounterparts(so ..."
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Abstract. It had been known since old times [MO, CL, Da] that there exists a universal finite type invariant (“an expansion”) Z old for Knotted Trivalent Graphs (KTGs), and that it can be chosen to intertwine between some of the standard operations on KTGs and their chorddiagrammaticcounterparts(sothatrelativetothoseoperations,itis“homomorphic”). YetperhapsthemostimportantoperationonKTGsisthe“edgeunzip”operation,andwhile the behavior of Z old under edge unzip is well understood, it is not plainly homomorphic as some “correction factors ” appear. In this paper we present two (equivalent) ways of modifying Z old into a new expansion Z, defined on “dotted Knotted Trivalent Graphs ” (dKTGs), which is homomorphic with respect to a large set of operations. The first is to replace “edge unzips ” by “tree connected sums”, and the second involves somewhat restricting the circumstances under which edge unzips are allowed. As we shall explain, the newly defined class dKTG of knotted trivalent graphs retains all the good qualities that KTGs have — it remains firmly connected with the Drinfel’d theory of associators and it is sufficiently rich to serve as a foundation for an “Algebraic Knot Theory”. As a further application, we present a simple proof of the good
FINITE TYPE INVARIANTS
, 2008
"... This is an overview article on finite type invariants, written for the Encyclopedia of Mathematical Physics. ..."
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This is an overview article on finite type invariants, written for the Encyclopedia of Mathematical Physics.