Double-partition Quantum Cluster Algebras

Quantum cluster algebras and quantum nilpotent algebras.

A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the correspondin...

Optimizing Teleportation Cost in Multi-Partition Distributed Quantum Circuits

There are many obstacles in quantum circuits implementation with large scales, so distributed quantum systems are appropriate solution for these quantum circuits. Therefore, reducing the number of quantum teleportation leads to improve the cost of implementing a quantum circuit. The minimum number of teleportations can be considered as a measure of the efficiency of distributed quantum systems....

Green Formula in Hall Algebras and Cluster Algebras

The objective of the present paper is to give a survey of recent progress on applications of the approaches of Ringel-Hall type algebras to quantum groups and cluster algebras via various forms of Green’s formula. In this paper, three forms of Green’s formula are highlighted, (1) the original form of Green’s formula [Gre][Rin2], (2) the degeneration form of Green’s formula [DXX] and (3) the pro...

Graded Quantum Cluster Algebras of Infinite Rank as Colimits

We provide a graded and quantum version of the category of rooted cluster algebras introduced by Assem, Dupont and Schiffler and show that every graded quantum cluster algebra of infinite rank can be written as a colimit of graded quantum cluster algebras of finite rank. As an application, for each k we construct a graded quantum infinite Grassmannian admitting a cluster algebra structure, exte...

Reiter’s Properties for the Actions of Locally Compact Quantum Goups on von Neumann Algebras