Compositio Mathematica

On Kontsevich's Hochschild cohomology conjecture

Po Hu a1, Igor Kriz a2 and Alexander A. Voronov a3
a1 Department of Mathematics, Wayne State University, 656 W. Kirby, Rm. 1150 Faculty/Administration Building, Detroit, MI 48202, USA
a2 Department of Mathematics, University of Michigan, 2074 E Hall, 525 E University Avenue, Ann Arbor, MI 48109-1109, USA
a3 Department of Mathematics, University of Minnesota, 127 Vincent, 206 Church Street SE, Minneapolis, MN 55455-0487, USA

Article author query
hu p   [Google Scholar] 
kriz i   [Google Scholar] 
voronov a   [Google Scholar] 


Generalizing a conjecture of Deligne, Kontsevich proposed that there should be a notion of Hochschild cohomology of algebras over the little cube operad (or its chain complex) which in a natural way generalizes Hochschild cohomology of associative algebras. He moreover conjectured that the Hochschild cohomology, in this new sense, of an algebra over the little k-cube operad is an algebra over the little (k + 1)-cube operad. In this paper, we precisely state and prove this conjecture.

(Published Online January 13 2006)
(Received March 28 2004)
(Accepted December 8 2004)

Key Words: operads; Hochschild cohomology; algebraic structures.

Maths Classification

18D50; 55P48; 13D03.