The shuffle bialgebra and the cohomology of commutative algebras. (English) Zbl 0728.13003

Gerstenhaber, M.; Schack, S. D.

J. Pure Appl. Algebra 70, No. 3, 263-272 (1991).

The authors show that a commutative bialgebra has a natural set of commuting endomorphisms which, when its product is replaced by the zero multiplication, trivially give it a \(\lambda\)-algebra structure. Applying this to the shuffle bialgebra, the authors obtain the Feigin-Tsigan and Loday \(\lambda\)-operations on the Hochschild cohomology \(H^{\bullet}(A,-)\) of a commutative algebra A. Also it is considered the Hodge decomposition of the Hochschild homology \(H_{\bullet}(A,-)\) and cohomology \(H^{\bullet}(A,-)\).

Reviewer: I.Crivei (Cluj-Napoca)

MSC:

13D03	(Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
13F50	Rings with straightening laws, Hodge algebras
16W30	Hopf algebras (associative rings and algebras) (MSC2000)

Keywords:

\(\lambda \) -algebra; Hodge decomposition of the Hochschild homology

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