In 1945 Gerhard Hochschild published "On the cohomology groups of an associative algebra" in the
Annals of Mathematics
and thereby created what is now called Hochschild theory. In 1963, Murray Gerstenhaber proved that the Hochschild cohomology of any associative algebra carries a super-Poisson algebra structure, comprised of a graded commutative cup product and an odd super Lie algebra structure that acts through graded derivations with respect to the product. Subsequently, a number of higher structures have been discovered, and a vast body of research concerning and/or using Hochschild theory has developed in many different fields in mathematics and physics.
Cite this article
Luchezar L. Avramov, Ragnar-Olaf Buchweitz, Wendy Lowen, Hochschild Cohomology in Algebra, Geometry, and Topology. Oberwolfach Rep. 13 (2016), no. 1, pp. 449–506DOI 10.4171/OWR/2016/10