Searcharxiv⌕ Search

arXiv subjects

Charles Torossian

Publications and source records attributed to Charles Torossian.

13 recordsLinked to original sources
arXiv2014

Logarithms and deformation quantization

We prove the statement/conjecture of M. Kontsevich on the existence of the logarithmic formality morphism. This question was open since 1999, and the main obstacle was the presence of $dr/r$ type singularities near the boundary $r=0$ in the integrals over compactified configuration spaces. The novelty of our approach is the use of local torus actions on configuration spaces of points in the upper half-plane. It gives rise to a version of Stokes' formula for differential forms with singularities at the boundary which implies the formality property. We also show that the logarithmic formality morphism admits a globalization from $\mathbb{R}^d$ to an arbitrary smooth manifold.

math.QA↗
arXiv2012

Biquantization of symmetric pairs and the quantum shift

The biquantization of symmetric pairs was studied in Cattaneo and Torossian in terms of Kontsevich-like graphs. This paper, also in view of recent results in Calaque et al, amends a minor mistake that did not spoil the main results of the paper. The mistake consisted in ignoring a regular term in the boundary contribution of some propagators. On the other hand, its correction brings back the quantum shift, present in the approaches by the orbit method, that was otherwise puzzlingly missing. In addition a detailed comparison of the two, equivalent, ways of defining biquantization working on the upper half plane or on one quadrant is presented, as well as a more conceptual approach to biquantization and the due corrections of some results of Cattaneo and Torossian in view of the aforementioned correction by the quantum shift. Finally, we review the triquantization construction developed for the treatment of characters by taking into accounts the same boundary problem as for the biquantization.

math.QA↗
arXiv2009

On triviality of the Kashiwara-Vergne problem for quadratic Lie algebras

We show that the Kashiwara-Vergne (KV) problem for quadratic Lie algebras (that is, Lie algebras admitting an invariant scalar product) reduces to the problem of representing the Campbell-Hausdorff series in the form ln(e^xe^y)=x+y+[x,a(x,y)]+[y,b(x,y)], where a(x,y) and b(x,y) are Lie series in x and y. This observation explains the existence of explicit rational solutions of the quadratic KV problem (see M. Vergne, C.R.A.S. 329 (1999), no. 9, 767--772 and A. Alekseev, E. Meinrenken, C.R.A.S. 335 (2002), no. 9, 723--728 arXiv:math/0209346), whereas constructing an explicit rational solution of the full KV problem would probably require the knowledge of a rational Drinfeld associator. It also gives, in the case of quadratic Lie algebras, a direct proof of the Duflo theorem (implied by the KV problem).

math.QA↗
arXiv2008

Applications de la bi-quantification à la théorie de Lie

This article 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 and some extensions are relating discussions with my student. 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.

math.QA↗
arXiv2008

Quantification pour les paires symetriques et diagrammes de Kontsevich

In this article we use the expansion for biquantization described in Cattaneo-Felder [math.QA/0309180] for the case of symmetric spaces. We introduce a function of two variables $E(X,Y)$ for any symmetric pairs. This function has an expansion in terms of Kontsevich's diagrams. We recover most of the known results though in a more systematic way by using some elementary properties of this $E$ function. We prove that Cattaneo and Felder's star product coincides with Rouviere's for any symmetric pairs. We generalize some of Lichnerowicz's results for the commutativity of the algebra of invariant differential operators and solve a long standing problem posed by M. Duflo for the expression of invariant differential operators on any symmetric spaces in exponential coordinates. We describe the Harish-Chandra homomorphism in the case of symmetric spaces by using all these constructions. We develop a new method to construct characters for algebras of invariant differential operators. We apply these methods in the case of $σ$-stable polarizations.

math.RT↗
arXiv2008

Solution non universelle pour le problème KV-78

In 78' M. Kashiwara and Vergne conjectured some property on the Campbell-Hausdorff series in such way a trace formula is satisfied. They proposed an explicit solution in the case of solvable Lie algebras. In this note we prove that this "solvable solution" is not universal. Our method is based on computer calculation. Furthermore our programs prove up to degree 16, Drinfeld's Lie algebra $\mathfrak{grt}_1$ coincides with the Lie algebra $\hat{kv_2}$ defined in \cite{AT}.

math.GR↗
arXiv2008

The Kashiwara-Vergne conjecture and Drinfeld's associators

The Kashiwara-Vergne (KV) conjecture is a property of the Campbell-Hausdorff 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 KV_n, and an extension \hat{KV}_2 of the group KV_2. We show that the group \hat{KV}_2 contains the Grothendieck-Teichmueller group GRT as a subgroup, and that it acts freely and transitively on the set of solutions of the KV problem Sol(KV). Furthermore, we prove that Sol(KV) is isomorphic to a direct product of a line \k (\k being a field of characteristic zero) and the set of solutions of the pentagon equation with values in the group KV_3. The latter contains the set of Drinfeld's associators as a subset. As a by-product, we obtain a new proof of the Kashiwara-Vergne conjecture based on the Drinfeld's theorem on existence of associators.

math.QA↗
arXiv2007

The Kashiwara-Vergne conjecture

This is a Bourbaki's seminar text. We introduce the combinatorial Kashiwara-Vergne conjecture on the Baker-Campbell-Hausdorff serie. After recalling previous results and consequences, we explain the Alekseev-Meinrenken's proof [math.QA/0506499]

math.QA↗
arXiv2005

Deux resultats autour du theoreme de restriction de Chevalley

We use Dunkl's operators to give an elementary proof of the surjectivity in the Chevalley's restriction theorem. In the second part of this article we describe the image of the invariants by the restriction map in the case of Takiff algebras. We extend this result to Takiff symmetric spaces.

math.RT↗
arXiv2003

Cohomologie tangente et cup-produit pour la quantification de Kontsevich

On a flat manifold, M. Kontsevich's formality quasi-isomorphism is compatible with cup-products on tangent cohomology spaces, in the sense that its derivative at any formal Poisson 2-tensor induces an isomorphism of graded commutative algebras from Poisson cohomology space to Hochschild cohomology space relative to the associated deformed multiplication. We give here a detailed proof of this result, with signs and orientations precised.

math.QA↗
arXiv2002

Kashiwara-Vergne-Rouviere methods for symmetric spaces

This article follows our previous work on Campbell-Hausdorff formula. We study the case of symmetric spaces. We recover, by using a Kontsevich's deformation of the Baker-Campbell-Hausdorff formula, Rouviere's results on the convolution of invariant distributions, for solvable symmetric spaces and "very symmetric spaces".

math.QA↗
arXiv2001

Convolution of invariant distributions: Proof of the Kashiwara-Vergne conjecture

Consider the Kontsevich $\star$-product on the symmetric algebra of a finite dimensional Lie algebra $\mathfrak g$, regarded as the algebra of distributions with support 0 on $\mathfrak g$. In this paper, we extend this $\star$-product to distributions satisfying an appropriate support condition. As a consequence, we prove a long standing conjecture of Kashiwara-Vergne on the convolution of germs of invariant distributions on the Lie group $G$.

math.QA↗