Related papers: Matrix Factorizations and Representations of Quive…

We obtain a presentation of the t-deformed Grothendieck ring of a quantum loop algebra of Dynkin type A, D, E. Specializing t at the the square root of the cardinality of a finite field F, we obtain an isomorphism with the derived Hall…

Given a quiver, a fixed dimension vector, and a positive integer n, we construct a functor from the category of D-modules on the space of representations of the quiver to the category of modules over a corresponding Gan-Ginzburg algebra of…

We show that the category of finite-dimensional modules over the endomorphism algebra of a rigid object in a Hom-finite triangulated category is equivalent to the Gabriel-Zisman localisation of the category with respect to a certain class…

We prove that for an odd prime $p$, the derived category $\mathcal{D}(KU_{(p)})$ of the $p$-local complex periodic $K$-theory spectrum $KU_{(p)}$ is triangulated equivalent to the derived category of its homotopy ring $\pi_*KU_{(p)}$. This…

We show that the category of projective modules over a graded commutative ring admits a triangulation with respect to module suspension if and only if the ring is a finite product of graded fields and exterior algebras on one generator over…

We show that equivariant tilting modules over equivariant algebras induce equivalences of derived factorization categories. As an application, we show that the derived category of a noncommutative resolution of a linear section of a…

Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type ADE can be recovered from the data of the corresponding…

We study the representations of two types of pointed Hopf algebras: restricted two-parameter quantum groups, and the Drinfel'd doubles of rank one pointed Hopf algebras of nilpotent type. We study, in particular, under what conditions a…

We define mutation pair in a pseudo-triangulated category. We prove that under certain conditions, for a mutation pair in a pseudo-triangulated category, the corresponding quotient category carries a natural triangulated structure. This…

We discuss an interesting duality known to occur for certain complex reflection groups, namely the duality groups. Our main construction yields a concrete, representation theoretic realisation of this duality. This allows us to naturally…

In association with a finite dimensional algebra A of global dimension two, we consider the endomorphism algebra of A, viewed as an object in the triangulated hull of the orbit category of the bounded derived category, in the sense of…

For each valued quiver $Q$ of Dynkin type, we construct a valued ice quiver $\Delta_Q^2$. Let $G$ be a simple connected Lie group with Dynkin diagram the underlying valued graph of $Q$. The upper cluster algebra of $\Delta_Q^2$ is graded by…

In this paper, we prove the dg affinity of formal deformation algebroid stacks over complex smooth algebraic varieties. For that purpose, we introduce the triangulated category of formal deformation modules which are cohomologically…

A factorization formula for certain automorphisms of a Poisson algebra associated to a quiver is proved, which involves framed versions of moduli spaces of quiver representations. This factorization formula is related to wall-crossing…

Let $X$ be a smooth scheme with an action of an algebraic group $G$. We establish an equivalence of two categories related to the corresponding moment map $\mu : T^*X \to Lie(G)^*$ - the derived category of G-equivariant coherent sheaves on…

We prove that moduli spaces of meromorphic quadratic differentials with simple zeroes on compact Riemann surfaces can be identified with spaces of stability conditions on a class of CY3 triangulated categories defined using quivers with…

For a finite dimensional algebra $A$, we prove that the bounded homotopy category of projective $A$-modules and the bounded derived category of $A$-modules are dual to each other via certain categories of locally-finite cohomological…

We study fundamental group of the exchange graphs for the bounded derived category D(Q) of a Dynkin quiver Q and the finite-dimensional derived category D(\Gamma_N Q) of the Calabi-Yau-N Ginzburg algebra associated to Q. In the case of…

Representations of small quantum groups $u_q({\mathfrak{g}})$ at a root of unity and their extensions provide interesting tensor categories, that appear in different areas of algebra and mathematical physics. There is an ansatz by Lusztig…

We study matrix factorizations of a section W of a line bundle on an algebraic stack. We relate the corresponding derived category (the category of D-branes of type B in the Landau-Ginzburg model with potential W) with the singularity…