Related papers: Quantum cluster variables via canonical submodules
Differential calculi are obtained for quantum homogeneous spaces by extending Woronowicz' approach to the present context. Representation theoretical properties of the differential calculi are investigated. Connections on quantum…
Let C be the category of finite-dimensional representations of a quantum affine algebra of simply-laced type. We introduce certain monoidal subcategories C_l (l integer) of C and we study their Grothendieck rings using cluster algebras.
We show that finite index quantum subgroups of a discrete quantum group are induced from finite index quantum subgroups of the unimodularization. As an application, we classify all finite index quantum subgroups of free products of the…
We show that the total cohomology of the canonical bundle of a smooth projective variety, seen as a module over an exterior algebra, splits into a natural direct sum of submodules which are generated in degree zero and have a linear free…
This survey article is concerned with the modeling of the kinematical structure of quantum systems in an algebraic framework which eliminates certain conceptual and computational difficulties of the conventional approaches. Relying on the…
Let $Q$ be a finite acyclic valued quiver. We give the cluster multiplication formulas in the quantum cluster algebra of $Q$ with arbitrary coefficients, by applying certain quotients of derived Hall subalgebras of $Q$. These formulas can…
In the theory of generalized cluster algebras, we build the so-called cluster formula and $D$-matrix pattern. Then as applications, some fundamental conjectures of generalized cluster algebras are solved affirmatively.
This is a survey of what is known and/or conjectured about the prime and primitive spectra of quantum algebras, of quantized coordinate rings in particular. The topological structure of these spectra, their relations to classical affine…
Quantum cluster states and entangled state analyzers are essential to measurement-based quantum computing. We propose to generate a quantum cluster-state and to make multipartite entanglement analyzer by using noninteracting free electrons…
It has been proved in \cite{LS} that cluster variables in cluster algebras of every skew-symmetric cluster algebra are positive. We prove that any regular generalized cluster variable of an affine quiver is positive. As a corollary, we…
This article introduces a novel approach to perform the simulation of a single qubit quantum algorithm using laser beams. Leveraging the polarization states of photonic qubits, and inspired by variational quantum eigensolvers, we develop a…
We consider the possibility of performing linear optical quantum computation making use of extra photonic degrees of freedom. In particular we focus on the case where we use photons as quadbits. The basic 2-quadbit cluster state is a…
We investigate compact quantum group actions on unital $C^*$-algebras by analyzing invariant subsets and invariant states. In particular, we come up with the concept of compact quantum group orbits and use it to show that countable compact…
This note introduces the superunitary region of a cluster algebra, the subspace of the totally positive region on which each cluster variable is at least 1. Our main result is that the superunitary region of a finite type cluster algebra is…
We compute the Hochschild cohomology and homology of a class of quantum exterior algebras, with coefficients in twisted bimodules. As a result we obtain several interesting examples of the homological behavior of these algebras.
Generalized quantum cluster algebras introduced in [1] are quantum deformation of generalized cluster algebras of geometric types. In this paper, we prove that the Laurent phenomenon holds in these generalized quantum cluster algebras. We…
Dual canonical bases are expected to satisfy a certain (double) triangularity property by Leclerc's conjecture. We propose an analogous conjecture for common triangular bases of quantum cluster algebras. We show that a weaker form of the…
We construct level-0 modules of the quantum affine algebra $\Uq$, as the $q$-deformed version of the Lie algebra loop module construction. We give necessary and sufficient conditions for the modules to be irreducible. We construct the…
We review how the (quantum) cluster algebra naturally emerges in the study of four-dimensional $\mathcal{N}=2$ supersymmetric gauge theories.
We introduce a class of non-commutative algebras that carry a non-commutative (geometric) cluster structure which are generated by identical copies of generalized Weyl algebras. Equivalent conditions for the finiteness of the set of the…