Drinfeld associators, Braid groups and explicit solutions of the Kashiwara–Vergne equations

Alekseev, Anton; Enriquez, Benjamin; Torossian, Charles

doi:10.1007/s10240-010-0029-4

Cited by 30 publications

(88 citation statements)

References 15 publications

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“…The pentagon equation is the most important element of the Drinfeld's theory of associators. Our main technical result shows that solutions of equation (3) with values in the group KRV 0 3 admit an almost unique decomposition of the form (2), and the corresponding automorphism F is automatically a solution of the KV problem (and, in particular, has the property (1)). …”

Section: Introductionmentioning

confidence: 96%

The Kashiwara-Vergne conjecture and Drinfeld's associators

2012

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The Kashiwara-Vergne (KV) conjecture is a property of the CampbellHausdorff series put forward in 1978. It has been settled in the positive by E. Meinrenken and the first author in 2006. In this paper, we study the uniqueness issue for the KV problem. To this end, we introduce a family of infinite-dimensional groups KRV 0 n , and a group KRV2 which contains KRV 0 2 as a normal subgroup. We show that KRV2 also contains the Grothendieck-Teichmüller group GRT1 as a subgroup, and that it acts freely and transitively on the set of solutions of the KV problem SolKV. Furthermore, we prove that SolKV is isomorphic to a direct product of affine line A 1 and the set of solutions of the pentagon equation with values in the group KRV 0 3 . The latter contains the set of Drinfeld's associators as a subset. As a by-product of our construction, we obtain a new proof of the Kashiwara-Vergne conjecture based on the Drinfeld's theorem on existence of associators.

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Section: Introductionmentioning

confidence: 96%

The Kashiwara-Vergne conjecture and Drinfeld's associators

2012

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“…We shall content ourselves here with a very brief survey of their method and refer to [7] and [2] for full details. Let us state and explain one of the main results of [7].…”

Section: An Abstract Formulation Of the Kashiwara-vergne Problemmentioning

confidence: 99%

“…We refer to [7], [2] and the monograph [21] for detailed statements of (ii) and (iii). Drinfeld's proof exhibits an example of arising from the monodromy of the Knizhnik-Zamolodchikov di¤erential equation…”

Section: Drinfeld' S Associators and The Kashiwara-vergne Conjecturementioning

confidence: 99%

“…The automorphism F is obtained as F = exp f , where the homogeneous components of f 2 td 2 are determined inductively; a closed formula for F is given by Theorem 4 in [2]. From the property (X + Y + Z) = X + Y + Z it follows that the Lie series (X; Y ) := F 1 (X + Y ) satis…es the associativity (X; (Y; Z)) = ( (X; Y ); Z).…”

Section: Drinfeld' S Associators and The Kashiwara-vergne Conjecturementioning

confidence: 99%

“…The Kontsevich deformation quantization was again one of the major tools in this approach; see Torossian's exposition [57]. Then, in 2008 Alekseev and Torossian [7] (see also [2]) gave a di¤erent proof of the conjecture, reducing the problem to the existence theorem of Drinfeld's associators. A recent paper by Carlo Rossi [41] sheds light on the link between both proofs.…”

Section: Notesmentioning

confidence: 99%

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Symmetric Spaces and the Kashiwara-Vergne Method

Rouvière

2014

Lecture Notes in Mathematics

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CONTENTS AbstractThe Kashiwara-Vergne method reduces the proof of a deep result in analysis on a Lie group (transferring convolution of invariant distributions from the group to its Lie algebra, by means of the exponential mapping) to checking two formal Lie brackets identities linked to the Campbell-Hausdor¤ formula. First, we expound this method and a proof, for quadratic or solvable Lie algebras, of the Kashiwara-Vergne conjecture allowing to apply the method to those cases. We then extend it to a general symmetric space S = G=H. This leads to introduce a function e(X; Y ) of two tangent vectors X; Y at the origin of S, allowing to make explicit, in the exponential chart, G-invariant di¤erential operators of S, the structure of the algebra of all such operators, and an expansion of mean value operators and spherical functions. For Riemannian symmetric spaces of the noncompact type, otherwise well-known from the work of Harish-Chandra and Helgason, we compare this approach with the classical one. For rank one spaces (the hyperbolic spaces), we give an explicit formula for e(X; Y ).Finally, we explain a construction of e for a general symmetric space by means of Lie series linked to the Campbell-Hausdor¤ formula, in the spirit of the original Kashiwara-Vergne method. Proved from this construction, the main properties of e thus link the fundamental tools of H-invariant analysis on a symmetric space to its in…nitesimal structure.The results extend to line bundles over symmetric spaces. PrefaceLet G=K denote a Riemannian symmetric space of the noncompact type.« The action of the G-invariant di¤ erential operators on G=K on the radial functions on G=K is isomorphic with the action of certain di¤ erential operators with constant coe¢ cients. The isomorphism in question is used in Harish-Chandra's work on the Fourier analysis on G and is related to the Radon transform on G=K.For the case when G is complex a more direct isomorphism of this type is given, again as a consequence of results of HarishChandra.» This is a quote from Sigurður Helgason, Fundamental solutions of invariant di¤ erential operators on symmetric spaces, Amer. J. Maths. 86 (1964), p. 566, one of the …rst mathematical papers I have studied. As a consequence he showed that every (non-zero) G-invariant di¤erential operator D on G=K has a fundamental solution, whence the solvability of the di¤erential equation Du = f . Those results were fascinating to me, they still are and they fostered my interest in invariant di¤erential operators as well as Radon transforms. Yet I was dreaming simpler proofs could be given, without relying on HarishChandra's deep study of semisimple Lie groups... In the beginning of 1977 Michel Du ‡o [18] gave an analytic interpretation of the celebrated Du ‡o isomorphism he had exhibited six years before by means of algebraic constructions in an enveloping algebra. As a consequence he proved that every (non-zero) bi-invariant di¤erential operator on a Lie group has a local fundamental solution and is locally solvable. His work us...

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Galois Action on Knots II: Proalgebraic String Links and Knots

Furusho

2020

Springer Proceedings in Mathematics &Amp; Statistics

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We discuss an action of the Grothendieck-Teichmüller proalgebraic group on the linear span of proalgebraic tangles, oriented tangles completed by a filtration of Vassiliev. The action yields a motivic structure on tangles. We derive distinguished properties of the action particularly on proalgebraic string links and on proalgebraic knots which can not be observed in the action on proalgebraic braids. By exploiting the properties, we explicitly calculate the inverse image of the trivial (the chordless) chord diagram under the Kontsevich isomorphism.

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Drinfeld associators, Braid groups and explicit solutions of the Kashiwara–Vergne equations

Cited by 30 publications

References 15 publications

The Kashiwara-Vergne conjecture and Drinfeld's associators

The Kashiwara-Vergne conjecture and Drinfeld's associators

Symmetric Spaces and the Kashiwara-Vergne Method

Galois Action on Knots II: Proalgebraic String Links and Knots

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