Counting in Calabi--Yau categories, with applications to Hall algebras and knot polynomials
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909364330168320 |
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| author | Gorsky, Mikhail Haiden, Fabian |
| author_facet | Gorsky, Mikhail Haiden, Fabian |
| contents | We show that homotopy cardinality -- a priori ill-defined for many dg-categories, including all periodic ones -- has a reasonable definition for even-dimensional Calabi--Yau (evenCY) categories and their relative generalizations (under appropriate finiteness conditions).
As a first application we solve the problem of defining an intrinsic Hall algebra for degreewise finite pre-triangulated dg-categories in the case of oddCY categories. We compare this definition with Toën's derived Hall algebras (in case they are well-defined) and with other approaches based on extended Hall algebras and central reduction, including a construction of Hall algebras associated with Calabi--Yau triples of triangulated categories. For a category equivalent to the root category of a 1CY abelian category $\mathcal A$, the algebra is shown to be isomorphic to the Drinfeld double of the twisted Ringel--Hall algebra of $\mathcal A$, thus resolving in the Calabi--Yau case the long-standing problem of realizing the latter as a Hall algebra intrinsically defined for such a triangulated category.
Our second application is the proof of a conjecture of Ng--Rutherford--Shende--Sivek, which provides an intrinsic formula for the ruling polynomial of a Legendrian knot $L$, and its generalization to Legendrian tangles, in terms of the augmentation category of $L$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10154 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Counting in Calabi--Yau categories, with applications to Hall algebras and knot polynomials Gorsky, Mikhail Haiden, Fabian Quantum Algebra Category Theory Rings and Algebras Representation Theory Symplectic Geometry We show that homotopy cardinality -- a priori ill-defined for many dg-categories, including all periodic ones -- has a reasonable definition for even-dimensional Calabi--Yau (evenCY) categories and their relative generalizations (under appropriate finiteness conditions). As a first application we solve the problem of defining an intrinsic Hall algebra for degreewise finite pre-triangulated dg-categories in the case of oddCY categories. We compare this definition with Toën's derived Hall algebras (in case they are well-defined) and with other approaches based on extended Hall algebras and central reduction, including a construction of Hall algebras associated with Calabi--Yau triples of triangulated categories. For a category equivalent to the root category of a 1CY abelian category $\mathcal A$, the algebra is shown to be isomorphic to the Drinfeld double of the twisted Ringel--Hall algebra of $\mathcal A$, thus resolving in the Calabi--Yau case the long-standing problem of realizing the latter as a Hall algebra intrinsically defined for such a triangulated category. Our second application is the proof of a conjecture of Ng--Rutherford--Shende--Sivek, which provides an intrinsic formula for the ruling polynomial of a Legendrian knot $L$, and its generalization to Legendrian tangles, in terms of the augmentation category of $L$. |
| title | Counting in Calabi--Yau categories, with applications to Hall algebras and knot polynomials |
| topic | Quantum Algebra Category Theory Rings and Algebras Representation Theory Symplectic Geometry |
| url | https://arxiv.org/abs/2409.10154 |