Lie, associative and commutative quasi-isomorphism
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2019
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| _version_ | 1866913734952222720 |
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| author | Campos, Ricardo Petersen, Dan Robert-Nicoud, Daniel Wierstra, Felix |
| author_facet | Campos, Ricardo Petersen, Dan Robert-Nicoud, Daniel Wierstra, Felix |
| contents | Over a field of characteristic zero, we show that two commutative differential graded (dg) algebras are quasi-isomorphic if and only if they are quasi-isomorphic as associative dg algebras. This answers a folklore problem in rational homotopy theory, showing that the rational homotopy type of a space is determined by its associative dg algebra of rational cochains. We also show a Koszul dual statement, under an additional completeness hypothesis: two homotopy complete dg Lie algebras whose universal enveloping algebras are quasi-isomorphic as associative dg algebras must themselves be quasi-isomorphic. The latter result applies in particular to nilpotent Lie algebras (not differential graded), in which case it says that two nilpotent Lie algebras whose universal enveloping algebras are isomorphic as associative algebras must be isomorphic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1904_03585 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Lie, associative and commutative quasi-isomorphism Campos, Ricardo Petersen, Dan Robert-Nicoud, Daniel Wierstra, Felix Rings and Algebras Algebraic Topology K-Theory and Homology Quantum Algebra Representation Theory Over a field of characteristic zero, we show that two commutative differential graded (dg) algebras are quasi-isomorphic if and only if they are quasi-isomorphic as associative dg algebras. This answers a folklore problem in rational homotopy theory, showing that the rational homotopy type of a space is determined by its associative dg algebra of rational cochains. We also show a Koszul dual statement, under an additional completeness hypothesis: two homotopy complete dg Lie algebras whose universal enveloping algebras are quasi-isomorphic as associative dg algebras must themselves be quasi-isomorphic. The latter result applies in particular to nilpotent Lie algebras (not differential graded), in which case it says that two nilpotent Lie algebras whose universal enveloping algebras are isomorphic as associative algebras must be isomorphic. |
| title | Lie, associative and commutative quasi-isomorphism |
| topic | Rings and Algebras Algebraic Topology K-Theory and Homology Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/1904.03585 |