Koszul duality and Calabi-Yau structures
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908338914066432 |
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| author | Holstein, Julian Rivera, Manuel |
| author_facet | Holstein, Julian Rivera, Manuel |
| contents | We show that Koszul duality between differential graded categories and pointed curved coalgebras interchanges smooth and proper Calabi-Yau structures. This result is a generalization and conceptual explanation of the following two applications. For a finite-dimensional Lie algebra a smooth Calabi-Yau structure on the universal enveloping algebra is equivalent to a proper Calabi-Yau structure on the Chevalley-Eilenberg chain coalgebra, which exists if and only if Poincare duality is satisfied. For a topological space X having the homotopy type of a finite complex we show an oriented Poincare duality structure (with local coefficients) on X is equivalent to a proper Calabi-Yau structure on the dg coalgebra of chains on X and to a smooth Calabi-Yau structure on the dg algebra of chains on the based loop space of X. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_03604 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Koszul duality and Calabi-Yau structures Holstein, Julian Rivera, Manuel Algebraic Topology Category Theory Quantum Algebra Representation Theory We show that Koszul duality between differential graded categories and pointed curved coalgebras interchanges smooth and proper Calabi-Yau structures. This result is a generalization and conceptual explanation of the following two applications. For a finite-dimensional Lie algebra a smooth Calabi-Yau structure on the universal enveloping algebra is equivalent to a proper Calabi-Yau structure on the Chevalley-Eilenberg chain coalgebra, which exists if and only if Poincare duality is satisfied. For a topological space X having the homotopy type of a finite complex we show an oriented Poincare duality structure (with local coefficients) on X is equivalent to a proper Calabi-Yau structure on the dg coalgebra of chains on X and to a smooth Calabi-Yau structure on the dg algebra of chains on the based loop space of X. |
| title | Koszul duality and Calabi-Yau structures |
| topic | Algebraic Topology Category Theory Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2410.03604 |