×

Homotopy Gerstenhaber algebras. (English) Zbl 0974.16005

Dito, Giuseppe (ed.) et al., Conférence Moshé Flato 1999: Quantization, deformations, and symmetries, Dijon, France, September 5-8, 1999. Volume II. Dordrecht: Kluwer Academic Publishers. Math. Phys. Stud. 22, 307-331 (2000).
A Gerstenhaber algebra (G-algebra) is defined by two operations, a (dot) product \(ab\) and a bracket \([a,b]\), on a graded vector space \(V\) over a field of characteristic zero. The bracket must be a graded derivation of the product in the following sense: \([a,bc]=[a,b]c+(-1)^{(|a|-1)|b|}b[a,c]\), where \(|a|\) denotes the degree of \(a\in V\). Hence a G-algebra is a specific graded version of a Poisson algebra, it may be equivalently defined as an algebra over the operad \(e_2=H_\bullet(D)=H_\bullet(D;k)\) of the homology of the little disks operad. P. Deligne gave the following conjecture: The structure of an algebra over the homology little disks operad \(e_2\) on the Hochschild cohomology may be naturally lifted to the (co)chain level. The conjecture has found several interpretations, both algebraic and geometric. The main purpose of this paper is to complete Getzler-Jones’ proof of the conjecture, thereby establishing an explicit relationship between the geometry of configurations of points in the plane and the Hochschild complex of an associative algebra.
For the entire collection see [Zbl 0949.00040].

MSC:

16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
55P48 Loop space machines and operads in algebraic topology
17A30 Nonassociative algebras satisfying other identities
18D50 Operads (MSC2010)
19D55 \(K\)-theory and homology; cyclic homology and cohomology
14H10 Families, moduli of curves (algebraic)
PDF BibTeX XML Cite
Full Text: arXiv