Crane and Frenkel proposed a notion of a Hopf category [2]. It was motivated by Lusztig’s approach to quantum groups – his theory of canonical bases. In particular, Lusztig obtains braided deformations Uqn+ of universal enveloping algebras Un+ for some nilpotent Lie algebras n+ together with canonical bases of these braided Hopf algebras [4, 5, 6]. The elements of the canonical basis are identi...

Crane and Frenkel proposed a notion of a Hopf category [2]. It was motivated by Lusztig’s approach to quantum groups – his theory of canonical bases. In particular, Lusztig obtains braided deformations Uqn+ of universal enveloping algebras Un+ for some nilpotent Lie algebras n+ together with canonical bases of these braided Hopf algebras [4, 5, 6]. The elements of the canonical basis are identi...

Abstract We introduce the notion of a rank function on triangulated category

Tropical friezes are the tropical analogues of Coxeter-Conway frieze patterns. In this note, we study them using triangulated categories. A tropical frieze on a 2-Calabi-Yau triangulated category C is a function satisfying a certain addition formula. We show that when C is the cluster category of a Dynkin quiver, the tropical friezes on C are in bijection with the n-tuples in Z, any tropical fr...

This paper introduces a mathematical definition of the category of D-branes in Landau-Ginzburg orbifolds in terms of A∞-categories. Our categories coincide with the categories of (gradable) matrix factorizations for quasi-homogeneous polynomials. After setting up the necessary definitions, we prove that our category for the polynomial x is equivalent to the derived category of representations o...

We prove for a large family of rings R that their λ-pure global dimension is greater than one for each infinite regular cardinal λ. This answers in negative a problem posed by Rosický. The derived categories of such rings then do not satisfy the Adams λ-representability for morphisms for any λ. Equivalently, they are examples of well generated triangulated categories whose λ-abelianization in t...

We introduce continuous Frobenius categories. These are topological categories which are constructed using representations of the circle over a discrete valuation ring. We show that they are Krull-Schmidt with one indecomposable object for each pair of (not necessarily distinct) points on the circle. By putting restrictions on these points we obtain various Frobenius subcategories. The main pur...

Using techniques due to Dwyer–Greenlees–Iyengar we construct weight structures in triangulated categories generated by compact objects. We apply our result to show that, for a dg category whose homology vanishes in negative degrees and is semi-simple in degree 0, each simple module over the homology lifts to a dg module which is unique up to isomorphism in the derived category. This allows us, ...

It is shown that the derived dimension of any representation-finite Artin algebra is at most one. Mathematics Subject Classification 2000: 16G10, 18E30, 18E10. Let T be a triangulated category, and I and J two full subcategories of T . Denote by 〈I〉 the smallest full subcategory of T containing I and closed under shifts, finite direct sums and direct summands. Denote by I∗J the full subcategory...

We compute a lower bound for the dimension of a triangulated category, given that a certain finite generation hypothesis holds for cohomology. The bound is given in terms of the complexity of pairs of objects. We then apply the theory to the stable derived categories of complete intersections and of certain Artin algebras, and obtain lower bounds for the representation dimensions of the latter.