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\(A_ \infty\) algebras and the cohomology of moduli spaces. (English) Zbl 0863.17017

Gindikin, S. G. (ed.) et al., Lie groups and Lie algebras: E. B. Dynkin’s seminar. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 169, 91-107 (1995).
First Hochschild cohomology is reviewed (with a misprint in the definition) and its relation to deformation of associative algebras. Then cyclic cohomology is reviewed and its relation to deformations of associative algebras which keep invariant a given inner product. Then the notion of a strongly homotopy associative algebra of an \(A_\infty\) algebra is defined, which goes back to [J. Stasheff, Trans. Am. Math. Soc. 108, 275-292, 293-312 (1963; Zbl 0114.39402)]. Hochschild cohomology for an \(A_\infty\) algebra is defined and its relation to deformations are given. Cyclic homology of an \(A_\infty\) algebra is defined and its relation to deformations which keep invariant a given inner product. Finally this is applied to a construction of M. Kontsevitch [Feynman diagrams and low dimensional topology, Prog. Math. 120, 97-121 (1994)] who associated functions and cycles on the space \({\mathcal M}_{g,n}\) of complex surfaces of genus \(g\) with \(n\) punctures to associative \(A_\infty\) algebras.
For the entire collection see [Zbl 0828.00011].
Reviewer: P.Michor (Wien)

MSC:

17B56 Cohomology of Lie (super)algebras
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
14J15 Moduli, classification: analytic theory; relations with modular forms
19D55 \(K\)-theory and homology; cyclic homology and cohomology
58D27 Moduli problems for differential geometric structures

Citations:

Zbl 0114.39402
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