Relative Calabi-Yau structures and ice quivers with potential
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916857351503872 |
|---|---|
| author | Keller, Bernhard Liu, Junyang |
| author_facet | Keller, Bernhard Liu, Junyang |
| contents | In 2015, Van den Bergh showed that complete 3-Calabi-Yau algebras over an algebraically closed field of characteristic 0 are equivalent to Ginzburg dg algebras associated with quivers with potential. He also proved the natural generalisation to higher dimensions and non-algebraically closed ground fields. The relative version of the notion of Ginzburg dg algebra is that of Ginzburg morphism. For example, every ice quiver with potential gives rise to a Ginzburg morphism. We generalise Van den Bergh's theorem by showing that, under suitable assumptions, any morphism with a relative Calabi-Yau structure is equivalent to a Ginzburg(-Lazaroiu) morphism. In particular, in dimension 3 and over an algebraically closed ground field of characteristic 0, it is given by an ice quiver with potential. Thanks to the work of Bozec-Calaque-Scherotzke, this result can also be viewed as a non-commutative analogue of Joyce-Safronov's Lagrangian neighbourhood theorem in derived symplectic geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_16222 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Relative Calabi-Yau structures and ice quivers with potential Keller, Bernhard Liu, Junyang Representation Theory Algebraic Geometry K-Theory and Homology Rings and Algebras Symplectic Geometry 16E45 In 2015, Van den Bergh showed that complete 3-Calabi-Yau algebras over an algebraically closed field of characteristic 0 are equivalent to Ginzburg dg algebras associated with quivers with potential. He also proved the natural generalisation to higher dimensions and non-algebraically closed ground fields. The relative version of the notion of Ginzburg dg algebra is that of Ginzburg morphism. For example, every ice quiver with potential gives rise to a Ginzburg morphism. We generalise Van den Bergh's theorem by showing that, under suitable assumptions, any morphism with a relative Calabi-Yau structure is equivalent to a Ginzburg(-Lazaroiu) morphism. In particular, in dimension 3 and over an algebraically closed ground field of characteristic 0, it is given by an ice quiver with potential. Thanks to the work of Bozec-Calaque-Scherotzke, this result can also be viewed as a non-commutative analogue of Joyce-Safronov's Lagrangian neighbourhood theorem in derived symplectic geometry. |
| title | Relative Calabi-Yau structures and ice quivers with potential |
| topic | Representation Theory Algebraic Geometry K-Theory and Homology Rings and Algebras Symplectic Geometry 16E45 |
| url | https://arxiv.org/abs/2307.16222 |