The shuffle bialgebra and the cohomology of commutative algebras (Q804644)

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 \(H^{\bullet}(A,-)\).