Related papers: The Hall algebra of a spherical object
In this paper we describe the twisted Hall algebra of bound quiver with small homological dimension. The description is given in the terms of the quadratic form associated with the corresponding bound quiver.
This is a survey on spherical Hopf algebras. We give criteria to decide when a Hopf algebra is spherical and collect examples. We discuss tilting modules as a mean to obtain a fusion subcategory of the non-degenerate quotient of the…
We construct a geometric system from which the Hall algebra can be recovered. This system inherently satisfies higher associativity conditions and thus leads to a categorification of the Hall algebra. We then suggest how to use this…
We give an explicit construction of the cusp eigenforms on an elliptic curve defined over a finite field using the theory of Hall algebras and the Langlands correspondence for function fields and $\GL_n$. As a consequence we obtain a…
Let $\mathcal {A}$ be a finitary hereditary abelian category. In this note, we use the associativity of the derived Hall algebra associated to the bounded derived category of $\mathcal {A}$, whose multiplication structure constants are…
Under a mild condition, the perfect derived category and the finite-dimensional derived category of a graded gentle one-cycle algebra are described as twisted root categories of certain infinite quivers of type $\mathbb{A}_\infty^\infty$.…
We study the derived Hall algebra of the partially wrapped Fukaya category of a surface. We give an explicit description of the Hall algebra for the disk with m marked intervals and we give a conjectural description of the Hall algebras of…
For a smooth surface $S$, Porta-Sala defined a categorical Hall algebra generalizing previous work in K-theory of Zhao and Kapranov-Vasserot. We construct semi-orthogonal decompositions for categorical Hall algebras of points on $S$. We…
Under suitably nice conditions, given a coalgebra object in a tensor category we compute the layers of its coradical (socle) filtration.
Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, Dugger-Shipley, ..., Toen and…
The most commonly known triangulated categories arise from chain complexes in an abelian category by passing to chain homotopy classes or inverting quasi-isomorphisms. Such examples are called `algebraic' because they originate from abelian…
We discuss some basic properties of the graded center of a triangulated category and compute examples arising in representation theory of finite dimensional algebras.
This article provides an overview of the techniques related to classification of spherical and more general objects within triangulated categories, and its relationship with algebraic geometry, representation theory and symplectic geometry.…
The goal of this note is to spell out the (apparently well-known and intuitively clear) notion of abelian category over an algebraic stack. In the future we will discuss the (much less evident) notion, when instead of an abelian category…
We give a geometric formulation of To\"en's derived Hall algebra by constructing Grothendieck's six operations for the derived category of lisse-\'etale constructible sheaves on the derived stacks of complexes. Our formulation is based on…
Given a complete Heyting algebra we construct an algebraic tensor triangulated category whose Bousfield lattice is the Booleanization of the given Heyting algebra. As a consequence we deduce that any complete Boolean algebra is the…
In this article we describe the triangulated structure of the bounded derived category of a gentle algebra by describing the triangles induced by the morphisms between indecomposable objects in a basis of their Hom-space.
By using the approach in \cite{XX2006} to Hall algebras arising in homologically finite triangulated categories, we find an `almost' associative multiplication structure for indecomposable objects in a 2-periodic triangulated category. As…
This paper has been withdrawn and replaced by arXiv:1309.5035. In this paper we describe some examples of so called spherical functors between triangulated categories, which generalize the notion of a spherical object. We also give…
The concept of a morphism determined by an object provides a method to construct or classify morphisms in a fixed category. We show that this works particularly well for triangulated categories having Serre duality. Another application of…