  Let $C$ be a differential graded coalgebra, $
  \bar\Omega C$ the Adams cobar construction and $C^\vee$ the dual algebra. We
prove that for a large class of coalgebras $C$ there is a natural isomorphism
of Gerstenhaber algebras between the Hochschild cohomologies $HH^\ast (C^\vee,
C ^\vee)$ and $HH^\ast (\bar\Omega C ; \bar\Omega C)$. This result permits to
describe a Hodge decomposition of the loop space homology of a closed oriented
manifold, in the sense of Chas-Sullivan, when the field of coefficients is of
characteristic zero.
