Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

Lie, associative and commutative quasi-isomorphism

In this talk I shall explain that two commutative differential graded (dg) algebras can be connected by a zig-zag of quasi-isomorphisms of commutative algebras if and only if they can be connected by a zig-zag of quasi-isomorphisms of associative algebras. I shall then explain that this implies the Koszul dual statement, which says that two dg Lie algebras can be connected by a zig-zag of quasi-isomorphisms if and only if their universal enveloping algebras can be connected by a zig-zag of associative algebras. A corollary of these results is that two (non-dg) Lie algebras are isomorphic if and only if their universal enveloping algebras are isomorphic as associative algebras. This is joint work with Ricardo Campos, Dan Petersen and Daniel Robert-Nicoud.