The Kashiwara-Vergne conjecture and Drinfeld's associators

Alekseev, Anton; Torossian, Charles

doi:10.4007/annals.2012.175.2.1

Cited by 74 publications

(199 citation statements)

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“…We shall now give a precise statement of the Kashiwara-Vergne « conjecture» -henceforth a theorem, now completely proved by two di¤erent methods in [5] and [7].…”

Section: The Kashiwara-vergne Conjecturementioning

confidence: 99%

“…In their 2009 note [8], using the free Lie algebra formalism they had developed in [7], Alekseev and Torossian discovered a remarkable link between equations (KV1) and (KV2). In this section we extract from their work the necessary material, eventually leading in 1.9 to a proof of the conjecture in two important cases.…”

Section: From (Kv1) Towards (Kv2)mentioning

confidence: 99%

“…The conjecture is now a theorem, proved in full generality by two di¤erent methods in [5] resp. [7], by Alekseev and Meinrenken resp. Alekseev and Torossian; see the Notes of Chapter 1 for more details.…”

Section: Introductionmentioning

confidence: 99%

“…solvable, following [8], resp. [48]) and a brief overview of the latest proof by Alekseev and Torossian [7] for the general case.…”

Section: Introductionmentioning

confidence: 99%

“…solvable, following [8], resp. [48]) and a brief overview of the latest proof by Alekseev and Torossian [7] for the general case.Apart from Chapter 1 the whole paper focuses on a convolution transfer formula from a symmetric space S = G=H to its tangent space s at the origin. Working on suitable neighborhoods of the origin, we prove the existence of a (non-unique) function e on s s such that, for all H-invariant distributions u; v on s and all test functions f on s,…”

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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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“…We shall now give a precise statement of the Kashiwara-Vergne « conjecture» -henceforth a theorem, now completely proved by two di¤erent methods in [5] and [7].…”

Section: The Kashiwara-vergne Conjecturementioning

confidence: 99%

Section: From (Kv1) Towards (Kv2)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…solvable, following [8], resp. [48]) and a brief overview of the latest proof by Alekseev and Torossian [7] for the general case.…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Symmetric Spaces and the Kashiwara-Vergne Method

Rouvière

2014

Lecture Notes in Mathematics

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Applications de la bi-quantification à la théorie de Lie

Torossian¹

2010

Higher Structures in Geometry and Physics

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This article in French, with a large English introduction, is a survey about applications of bi-quantization theory in Lie theory. We focus on a conjecture of M. Duflo. Most of the applications are coming from our article with Alberto Cattaneo [13] and some extensions are relating discussions with my student [9]. The end of the article is completely new. We prove that the conjecture E = 1 implies the Kashiwara-Vergne conjecture. Our deformation is non geometric but uses a polynomial deformation of the coefficients.

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Computational Aspects of Some Exponential Identities

Casas

2018

Springer Proceedings in Mathematics &Amp; Statistics

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The Kashiwara-Vergne conjecture and Drinfeld's associators

Cited by 74 publications

References 19 publications

Symmetric Spaces and the Kashiwara-Vergne Method

Symmetric Spaces and the Kashiwara-Vergne Method

Applications de la bi-quantification à la théorie de Lie

Computational Aspects of Some Exponential Identities

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