Gerstenhaber bracket on double Hochschild complex and deformation theory

Abstract: We construct a differential and a Lie bracket on the spacelinebreak

H o m (A^{o t i m e s k}, A^{o t i m e s l}),_{k, l g e 0}

for any associative algebra

A

. The restriction of this bracket to the space

H o m (A^{o t i m e s k}, A),_{k g e 0}

is exactly the Gerstenhaber bracket. We discuss some formality conjecture related with this construction. We also discuss some applications to deformation theory.

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