# Duality between Lie and commutative algebras, revisited

SPEAKER: Dev Sinha

TITLE: Duality between Lie and commutative algebras, revisited.

ABSTRACT Almost forty years ago Quillen established a strong connection between (graded, differential) Lie and commutative co-algebras, giving an adjoint pair of functors between these categories which preserved their “model structures.” In one direction, one starts with a Lie algebra and obtains an algebra as the cohomology of its universal enveloping algebra. In the other direction, one can take primitives in a free-coalgebra replacement. Our main result is completing this story by giving a linearly dual pair of functors between Lie co-algebras and commutative algebras. The functor from commutative algebras to Lie co-algebras has a pleasant description in terms of graph homology, in the spirit of Kontsevich. This story might (or might not) be useful in defining functionals on Lie algebras.