University of Cambridge > Talks.cam > Differential Geometry and Topology Seminar > Lie, associative and commutative quasi-isomorphism

Lie, associative and commutative quasi-isomorphism

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  • UserDan Petersen, Stockholm World_link
  • ClockWednesday 27 February 2019, 16:00-17:00
  • HouseMR13.

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We present two “Koszul dual” theorems: (A) If two commutative dg algebras in characteristic zero are quasi-isomorphic as dg algebras, then they are also quasi-isomorphic as commutative dg algebras. (B) If two dg Lie algebras in characteristic zero have universal enveloping algebras which are quasi-isomorphic as dg algebras, then the dg Lie algebras are themselves quasi-isomorphic. Theorem B says in particular that two Lie algebras in characteristic zero (with no grading or differential) are isomorphic if and only if their universal enveloping algebras are isomorphic as algebras; even this result is new and answers an open question. Via the work of Quillen and Sullivan, these theorems have immediate consequences in rational homotopy theory: two simply connected spaces have the same rational homotopy type if and only if their algebras of rational cochains are quasi-isomorphic, if and only if the algebras of rational chains on their Moore loop spaces are quasi-isomorphic. (Joint work with R. Campos, D. Robert-Nicoud, F. Wierstra)

This talk is part of the Differential Geometry and Topology Seminar series.

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