Lie, associative and commutative quasi-isomorphism

Abstract: Let A and A' be commutative dg algebras over Q. There are two a priori different notions of what it means for them to be quasi-isomorphic: one could ask for a zig-zag of quasi-isomorphisms in the category of commutative dg algebras, or a zig-zag in the larger category of not necessarily commutative dg algebras. Our first main result is that these two notions coincide. The second main result is Koszul dual to the first, and states that if two dg Lie algebras over Q have quasi-isomorphic universal enveloping algebras, then the derived completions of the two dg Lie algebras are quasi-isomorphic. The latter result is new even for classical Lie algebras concentrated in degree zero. Both results have immediate consequences in rational homotopy theory. (Joint with Campos, Robert-Nicoud, Wierstra)

algebraic topologycategory theoryquantum algebra

Audience: learners

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