Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry

Calaque, Damien; Rossi, Carlo A.

doi:10.4171/096

Cited by 36 publications

(58 citation statements)

References 30 publications

(45 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The main argument, of homotopical nature, was sketched by Kontsevich in [20], later clarified by Manchon and Torossian in [21] in the framework of deformation quantization, and finally adapted to the case of Q-manifolds in [3]. The globalisation of the compatibility between cup products was first seriously considered in [5], and is addressed in Section 8.…”

Section: For Any Graded Vector Space a The Brace Operations On B = Enmentioning

confidence: 99%

“…The second one is when the MCE γ is of polyvector degree at most 1; then one can prove that so is its image U(γ), which can be interpreted in terms of a Fedosov connection and its Weyl curvature on a deformed algebra, following the terminology of [10]. Finally, the third case of interest is when the MCE γ is precisely a vector field; we are able to compute explicitly the quasi-isomorphisms U γ,1 and S γ,0 by means of a rooted Todd class j(γ), following [5] (see also [3]). …”

Section: For Any Graded Vector Space a The Brace Operations On B = Enmentioning

confidence: 99%

“…Moreover, Kontsevich claimed and sketchily proved in [20,Section 8] (see [3,21] for detailed proofs in particular cases) that U γ,1 is compatible with cup-products in the sense that it induces an algebra isomorphism…”

Section: For Any Graded Vector Space a The Brace Operations On B = Enmentioning

confidence: 99%

“…C A , admit compactificationsà la Fulton-MacPherson, obtained by successive real blow-ups: we will not discuss here the construction of their compactifications C + A,B , C A , which are smooth manifolds with corners, referring to [20], [21], [3] for more details, but we focus mainly on their stratification, in particular on the boundary strata of codimension 1 of C + A,B . Namely, the compactified configuration space C + A,B is a stratified space, and its boundary strata of codimension 1 look like as follows:…”

Section: Configuration Spaces C +mentioning

confidence: 99%

“…We refer to [3,21] for an explicit proof of (33) and (34), which we omit here. Stokes' Theorem implies the following identity between weights:…”

Section: First Of All For a Non-negative Integer M We Definementioning

confidence: 99%

See 4 more Smart Citations

Compatibility with Cap-Products in Tsygan’s Formality and Homological Duflo Isomorphism

Calaque

Rossi

2010

Lett Math Phys

Self Cite

View full text Add to dashboard Cite

Abstract. In this paper we prove, with details and in full generality, that the isomorphism induced on tangent homology by the Shoikhet-Tsygan formality L∞-quasi-isomorphism for Hochschild chains is compatible with capproducts. This is a homological analog of the compatibility with cup-products of the isomorphism induced on tangent cohomology by Kontsevich formality L∞-quasi-isomorphism for Hochschild cochains.As in the cohomological situation our proof relies on a homotopy argument involving a variant of Kontsevich eye. In particular we clarify the rôle played by the I-cube introduced in [4].Since we treat here the case of a most possibly general Maurer-Cartan element, not forced to be a bidifferential operator, then we take this opportunity to recall the natural algebraic structures on the pair of Hochschild cochain and chain complexes of an A∞-algebra. In particular we prove that they naturally inherit the structure of an A∞-algebra with an A∞-(bi)module.

show abstract

Section: For Any Graded Vector Space a The Brace Operations On B = Enmentioning

confidence: 99%

Section: For Any Graded Vector Space a The Brace Operations On B = Enmentioning

confidence: 99%

Section: For Any Graded Vector Space a The Brace Operations On B = Enmentioning

confidence: 99%

Section: Configuration Spaces C +mentioning

confidence: 99%

“…We refer to [3,21] for an explicit proof of (33) and (34), which we omit here. Stokes' Theorem implies the following identity between weights:…”

Section: First Of All For a Non-negative Integer M We Definementioning

confidence: 99%

See 3 more Smart Citations

Compatibility with Cap-Products in Tsygan’s Formality and Homological Duflo Isomorphism

Calaque

Rossi

2010

Lett Math Phys

Self Cite

View full text Add to dashboard Cite

show abstract

From Atiyah Classes to Homotopy Leibniz Algebras

Chen

Stiénon

2015

Commun. Math. Phys.

105

View full text Add to dashboard Cite

A celebrated theorem of Kapranov states that the Atiyah class of the tangent bundle of a complex manifold $X$ makes $T_X[-1]$ into a Lie algebra object in $D^+(X)$, the bounded below derived category of coherent sheaves on $X$. Furthermore Kapranov proved that, for a K\"ahler manifold $X$, the Dolbeault resolution $\Omega^{\bullet-1}(T_X^{1,0})$ of $T_X[-1]$ is an $L_\infty$ algebra. In this paper, we prove that Kapranov's theorem holds in much wider generality for vector bundles over Lie pairs. Given a Lie pair $(L,A)$, i.e. a Lie algebroid $L$ together with a Lie subalgebroid $A$, we define the Atiyah class $\alpha_E$ of an $A$-module $E$ (relative to $L$) as the obstruction to the existence of an $A$-compatible $L$-connection on $E$. We prove that the Atiyah classes $\alpha_{L/A}$ and $\alpha_E$ respectively make $L/A[-1]$ and $E[-1]$ into a Lie algebra and a Lie algebra module in the bounded below derived category $D^+(\mathcal{A})$, where $\mathcal{A}$ is the abelian category of left $\mathcal{U}(A)$-modules and $\mathcal{U}(A)$ is the universal enveloping algebra of $A$. Moreover, we produce a homotopy Leibniz algebra and a homotopy Leibniz module stemming from the Atiyah classes of $L/A$ and $E$, and inducing the aforesaid Lie structures in $D^+(\mathcal{A})$.Comment: 36 page

show abstract

M. Kontsevich’s graph complex and the Grothendieck–Teichmüller Lie algebra

2014

View full text Add to dashboard Cite

We show that the zeroth cohomology of M. Kontsevich's graph complex is isomorphic to the Grothendieck-Teichmüller Lie algebra grt 1 . The map is explicitly described. This result has applications to deformation quantization and Duflo theory. We also compute the homotopy derivations of the Gerstenhaber operad. They are parameterized by grt 1 , up to one class (or two, depending on the definitions). More generally, the homotopy derivations of the (non-unital) En operads may be expressed through the cohomology of a suitable graph complex. Our methods also give a second proof of a result of H. Furusho, stating that the pentagon equation for grt 1 -elements implies the hexagon equation.Theorem 1.2. The automorphism of the Gerstenhaber operad up to homotopy are given by grt 1 and one class, i.e., there is an isomorphisms of Lie algebras H 0 (Der(hoe 2 )) ∼ = grt 1 ⋊ K =: grt where K acts on grt 1 by multiplication with the degree with respect to the grading on grt 1 .

show abstract

Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry

Cited by 36 publications

References 30 publications

Compatibility with Cap-Products in Tsygan’s Formality and Homological Duflo Isomorphism

Compatibility with Cap-Products in Tsygan’s Formality and Homological Duflo Isomorphism

From Atiyah Classes to Homotopy Leibniz Algebras

M. Kontsevich’s graph complex and the Grothendieck–Teichmüller Lie algebra

Contact Info

Product

Resources

About