Théorème de formalité pour les g-variétés

Translated title of the contribution: Formality theorem for g-manifolds

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

With any g-manifold M are associated two dglas tot(ΛgkTpoly (M)) and tot(ΛgkDpoly (M)), whose cohomologies HCE (g,Tpoly (M)→0Tpoly •+1(M)) and HCE (g,Dpoly (M)→dHDpoly •+1(M)) are Gerstenhaber algebras. We establish a formality theorem for g-manifolds: there exists an L quasi-isomorphism Φ:tot(ΛgkTpoly (M))→tot(ΛgkDpoly (M)) whose first ‘Taylor coefficient’ (1) is equal to the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd cocycle of the g-manifold M, and (2) induces an isomorphism of Gerstenhaber algebras on the level of cohomology. Consequently, the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd class of the g-manifold M is an isomorphism of Gerstenhaber algebras from HCE (g,Tpoly (M)→0Tpoly •+1(M)) to HCE (g,Dpoly (M)→dHDpoly •+1(M)).

Original languageFrench
Pages (from-to)582-589
Number of pages8
JournalComptes Rendus Mathematique
Volume355
Issue number5
DOIs
StatePublished - May 1 2017

Fingerprint

G-manifolds
Formality
Gerstenhaber Algebra
Isomorphism
Square root
Theorem
Cohomology
Cocycle
Coefficient

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

@article{248968f3769746d3ac91bf78788fd9b5,
title = "Th{\'e}or{\`e}me de formalit{\'e} pour les g-vari{\'e}t{\'e}s",
abstract = "With any g-manifold M are associated two dglas tot(Λ•g∨⊗kTpoly •(M)) and tot(Λ•g∨⊗kDpoly •(M)), whose cohomologies HCE •(g,Tpoly •(M)→0Tpoly •+1(M)) and HCE •(g,Dpoly •(M)→dHDpoly •+1(M)) are Gerstenhaber algebras. We establish a formality theorem for g-manifolds: there exists an L∞ quasi-isomorphism Φ:tot(Λ•g∨⊗kTpoly •(M))→tot(Λ•g∨⊗kDpoly •(M)) whose first ‘Taylor coefficient’ (1) is equal to the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd cocycle of the g-manifold M, and (2) induces an isomorphism of Gerstenhaber algebras on the level of cohomology. Consequently, the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd class of the g-manifold M is an isomorphism of Gerstenhaber algebras from HCE •(g,Tpoly •(M)→0Tpoly •+1(M)) to HCE •(g,Dpoly •(M)→dHDpoly •+1(M)).",
author = "Liao, {Hsuan Yi} and Stienon, {Mathieu Philippe} and Ping Xu",
year = "2017",
month = "5",
day = "1",
doi = "10.1016/j.crma.2017.03.008",
language = "French",
volume = "355",
pages = "582--589",
journal = "Comptes Rendus Mathematique",
issn = "1631-073X",
publisher = "Elsevier Masson",
number = "5",

}

Théorème de formalité pour les g-variétés. / Liao, Hsuan Yi; Stienon, Mathieu Philippe; Xu, Ping.

In: Comptes Rendus Mathematique, Vol. 355, No. 5, 01.05.2017, p. 582-589.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Théorème de formalité pour les g-variétés

AU - Liao, Hsuan Yi

AU - Stienon, Mathieu Philippe

AU - Xu, Ping

PY - 2017/5/1

Y1 - 2017/5/1

N2 - With any g-manifold M are associated two dglas tot(Λ•g∨⊗kTpoly •(M)) and tot(Λ•g∨⊗kDpoly •(M)), whose cohomologies HCE •(g,Tpoly •(M)→0Tpoly •+1(M)) and HCE •(g,Dpoly •(M)→dHDpoly •+1(M)) are Gerstenhaber algebras. We establish a formality theorem for g-manifolds: there exists an L∞ quasi-isomorphism Φ:tot(Λ•g∨⊗kTpoly •(M))→tot(Λ•g∨⊗kDpoly •(M)) whose first ‘Taylor coefficient’ (1) is equal to the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd cocycle of the g-manifold M, and (2) induces an isomorphism of Gerstenhaber algebras on the level of cohomology. Consequently, the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd class of the g-manifold M is an isomorphism of Gerstenhaber algebras from HCE •(g,Tpoly •(M)→0Tpoly •+1(M)) to HCE •(g,Dpoly •(M)→dHDpoly •+1(M)).

AB - With any g-manifold M are associated two dglas tot(Λ•g∨⊗kTpoly •(M)) and tot(Λ•g∨⊗kDpoly •(M)), whose cohomologies HCE •(g,Tpoly •(M)→0Tpoly •+1(M)) and HCE •(g,Dpoly •(M)→dHDpoly •+1(M)) are Gerstenhaber algebras. We establish a formality theorem for g-manifolds: there exists an L∞ quasi-isomorphism Φ:tot(Λ•g∨⊗kTpoly •(M))→tot(Λ•g∨⊗kDpoly •(M)) whose first ‘Taylor coefficient’ (1) is equal to the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd cocycle of the g-manifold M, and (2) induces an isomorphism of Gerstenhaber algebras on the level of cohomology. Consequently, the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd class of the g-manifold M is an isomorphism of Gerstenhaber algebras from HCE •(g,Tpoly •(M)→0Tpoly •+1(M)) to HCE •(g,Dpoly •(M)→dHDpoly •+1(M)).

UR - http://www.scopus.com/inward/record.url?scp=85016565036&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85016565036&partnerID=8YFLogxK

U2 - 10.1016/j.crma.2017.03.008

DO - 10.1016/j.crma.2017.03.008

M3 - Article

AN - SCOPUS:85016565036

VL - 355

SP - 582

EP - 589

JO - Comptes Rendus Mathematique

JF - Comptes Rendus Mathematique

SN - 1631-073X

IS - 5

ER -