Lie algebra structures on cohomology complexes of some H-pseudoalgebras

Abstract

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 (LP(V)=⨁n∈ZLPn(V),[,]), and prove that φ∈LP1 satisfies [φ,φ]=0 if and only if it determines a pseudoalgebra structure on V over the operad P, and LP(V) is a differential graded Lie algebra with a differentiation determined by the structure map φ of the H-pseudoalgebra V. Moreover, we construct a Lie algebra by using symmetric Schouten product for any pseudoalgebra V over the operad Lie.