2024/09/16 by Mikhail Gorsky, Gorsky, Mikhail, Fabian Haiden +1
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA) #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2409.10154
openalex publication_date 2024/09/16 · openalex created_date 2024/10/24 · openalex updated_date 2026/08/04
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.