The intrinsic bracket on the deformation complex of an associative algebra

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Abstract

In his pioneering work on deformation theory of associative algebras, Gerstenhaber created a bracket on the Hochschild cohomology Hoch(A,A), but this bracket seemed to be rather a tour de force since it was not induced from a differential graded Lie algebra structure on the underlying complex. Schlessinger and Stasheff constructed a differential graded Lie algebra structure on a complex giving the Harrison cohomology Harr(A,A) of a commutative algebra A in characteristic 0. Here we present a differential graded Lie algebra structure on a complex giving the Hochschild cohomology Hoch(A,A) and inducing the Gerstenhaber bracket for any associative algebra in any characteristic. Although the principal is the same as in the commutative case, the details as well as the essential idea will hopefully be revealed more transparently.

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