Lie algebra structures on cohomology complexes of some H-pseudoalgebras

Suppose H is a cocommutative Hopf algebra and P is the operad Ass, or Lie. For any left H-module V, we construct a graded Lie algebra (L-P(V) = (D-n is an element of Z L-P(n)(V), [,]), and prove that phi is an element of L-P(1) satisfies [phi, phi] = 0 if and only if it determines a pseudoalgebra structure on V over the operad P, and L-P(V) is a differential graded Lie algebra with a differentiation determined by the structure map phi of the H-pseudoalgebra V. Moreover, we construct a Lie algebra by using symmetric Schouten product for any pseudoalgebra V over the operad Lie. (C) 2013 Elsevier Inc. All rights reserved.