Related papers: Quantum Dilogarithm Identities at Root of Unity

We first study a new family of graded quiver varieties together with a new $t$-deformation of the associated Grothendieck rings. This provides the geometric foundations for a joint paper by Yoshiyuki Kimura and the author. We further…

Quantum groups at roots of unity have the property that their centre is enlarged. Polynomial equations relate the standard deformed Casimir operators and the new central elements. These relations are important from a physical point of view…

We give deterministic polynomial-time algorithms that, given an order, compute the primitive idempotents and determine a set of generators for the group of roots of unity in the order. Also, we show that the discrete logarithm problem in…

We compute the center and Azumaya locus in the simplest non-abelian examples of quantized multiplicative quiver varieties at a root of unity: quantum Weyl algebras of rank $N$, and quantum differential operators on the quantum group…

In this article the quantized matrix algebras as in the title have been studied at a root of unity. A full classification of simple modules over such quantized matrix algebras of rank $2$ along with a class of finite dimensional…

Let $Q$ be a finite quiver without oriented cycles and $k$ an algebraically closed field.In this paper we establish a connection between cluster algebras and the representation theory of the path algebra $kQ$, in terms of the spectral…

In \cite{rupel3},the authors defined algebra homomorphisms from the dual Ringel-Hall algebra of certain hereditary abelian category $\mathcal{A}$ to an appropriate $q$-polynomial algebra. In the case that $\mathcal{A}$ is the representation…

We study the restricted form of the qaunatized enveloping algebra of an untwisted affine Lie algebra and prove a triangular decomposition for it. In proving the decomposition we prove several new identities in the quantized algebra, one of…

Let $A$ be the path algebra of a finite acyclic quiver $Q$ over a finite field. We realize the quantum cluster algebra with principal coefficients associated to $Q$ as a sub-quotient of a certain Hall algebra involving the category of…

Here is a list of chapters: 1 Introduction 2 Notation and preliminaries Part I: Finite quantum groups 3 2x2 Matrix quantum groups and the quantum plane 4 Quantum enveloping algebras at a root of unity Part II: q-Oscillators 5…

We study the family of Y-systems and T-systems associated with the sine-Gordon models and the reduced sine-Gordon models for the parameter of continued fractions with two terms. We formulate these systems by cluster algebras, which turn out…

An important problem in the theory of cluster algebras is to compute the fundamental group of the exchange graph. A non-trivial closed loop in the exchange graph, for example, generates a non-trivial identity for the classical and quantum…

Quantum algebra of differential operators are studied

It this note we investigate the structure of the group of \sigma-unitary units in some noncommutative modular group algebras KG, where \sigma is a non-classical ring involution of KG.

We investigate the relation between Cartan decompositions of the unitary group and discrete quantum symmetries. To every Cartan decomposition there corresponds a quantum symmetry which is the identity when applied twice. As an application,…

We show that in case a cluster algebra coincides with its upper cluster algebra and the cluster algebra admits a grading with finite dimensional homogeneous components, the corresponding Berenstein-Zelevinsky quantum cluster algebra can be…

Let $(Q,W)$ be a quiver with a non degenerate potential. We give a new description of the \textbf{c}-vectors of $Q$. We use it to show that, if $Q$ is mutation equivalent to a Dynkin quiver, then the set of positive $\mathbf{c}$-vectors of…

Several Clifford algebras that are covariant under the action of a Lie algebra $g$ can be deformed in a way consistent with the deformation of $Ug$ into a quantum group (or into a triangular Hopf algebra) $U_qg$, i.e. so as to remain…

We classify up to isomorphism the quantum generalized Weyl algebras and determine their automorphism groups in all cases in a uniform way, including those where the parameter q is a root of unity, thereby completing the results obtained by…

We provide a complete classification of the singularities of cluster algebras of finite type with trivial coefficients. Alongside, we develop a constructive desingularization of these singularities via blowups in regular centers over fields…