Lie, associative and commutative quasi-isomorphism

• DOI: 10.48550/arxiv.1904.03585
• Publication Date: 2019-01-01
• Authors: Ricardo Campos; Dan Petersen; Daniel Robert-Nicoud; Felix Wierstra
• Keywords: Associative property; Isomorphism (crystallography); Associative algebra

Over a field of characteristic zero, we show that two commutative differential graded (dg) algebras are quasi-isomorphic if and only they as associative dg algebras. This answers folklore problem in rational homotopy theory, showing the type space is determined by its algebra cochains. We also Koszul dual statement, under an additional completeness hypothesis: complete Lie whose universal enveloping must themselves be quasi-isomorphic. The latter result applies particular to nilpotent (not graded), which case it says isomorphic isomorphic.