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).Comment: 8 page

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.Comment: in French, plain-TeX, 31 pages, 8 eps figures. Erreurs typographiques et signes corriges dans le theoreme 4.6 et la proposition 4.

Kontsevich Deformation Quantization and Flat Connections

In Torossian (J Lie Theory 12(2):597-616, 2002), the second author used the Kontsevich deformation quantization technique to define a natural connection ω n on the compactified configuration spaces $${\overline{C}_{n,0}}$$ of n points on the upper half-plane. Connections ω n take values in the Lie algebra of derivations of the free Lie algebra with n generators. In this paper, we show that ω n is flat. The configuration space $${\overline{C}_{n,0}}$$ contains a boundary stratum at infinity which coincides with the (compactified) configuration space of n points on the complex plane. When restricted to this stratum, ω n gives rise to a flat connection $${\omega_n^\infty}$$. We show that the parallel transport $${\Phi}$$ defined by the connection $${\omega_3^\infty}$$ between configuration 1(23) and (12)3 verifies axioms of an associator. We conjecture that $${\omega_n^\infty}$$ takes values in the Lie algebra $${\mathfrak{t}_n}$$ of infinitesimal braids. If correct, this conjecture implies that $${\Phi \in \exp(\mathfrak{t}_3)}$$ is a Drinfeld's associator. Furthermore, we prove $${\Phi \neq \Phi_{KZ}}$$ showing that $${\Phi}$$ is a new explicit solution of associator axiom

PBW for an inclusion of Lie algebras

Let h \subset g be an inclusion of Lie algebras with quotient h-module n. There is a natural degree filtration on the h-module U(g)/U(g)h whose associated graded h-module is isomorphic to S(n). We give a necessary and sufficient condition for the existence of a splitting of this filtration. In turn such a splitting yields an isomorphism between the h-modules U(g)/U(g)h and S(n). For the diagonal embedding h \subset h \oplus h the condition is automatically satisfied and we recover the classical Poincae-Birkhoff-Witt theorem. The main theorem and its proof are direct translations of results in algebraic geometry, obtained using an ad hoc dictionary. This suggests the existence of a unified framework allowing the simultaneous study of Lie algebras and of algebraic varieties, and a closely related work in this direction is on the way.Comment: Major revision, proofs of several results rewritten. Added a section explaining the case of a general representation, as opposed to the trivial one. 20 pages, LaTe

Solution non universelle pour le problème KV-78

Ajout dans section 3.2. Corrections typographiques dans équation (10) et bibliographie actualisée.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 $\widehat{kv_2}$ defined in \cite{AT}