Quantum Q systems: from cluster algebras to quantum current 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...

On Homomorphisms from Ringel-hall Algebras to Quantum Cluster Algebras

In [1],the authors defined algebra homomorphisms from the dual RingelHall algebra of certain hereditary abelian categoryA to an appropriate q-polynomial algebra. In the case that A is the representation category of an acyclic quiver, we give an alternative proof by using the cluster multiplication formulas in [9]. Moreover, if the underlying graph of Q is bipartite and the matrix B associated t...

`-restricted Q-systems and quantum affine algebras

Kuniba, Nakanishi, and Suzuki (1994) have formulated a general conjecture expressing the positive solution of an `-restricted Q-system in terms of quantum dimensions of Kirillov-Reshetikhin modules. After presenting this conjecture, we sketch a proof for the exceptional type E6 following our preprint (2013). In types E7 and E8, we prove positivity for a subset of the nodes of the Dynkin diagram...

Acyclic Quantum Cluster Algebras

This thesis concerns quantum cluster algebras. For skew-symmetric acyclic quantum cluster algebras, we express the quantum F -polynomials and the quantum cluster monomials in terms of Serre polynomials of quiver Grassmannians of rigid modules. Then we introduce a new family of graded quiver varieties together with a new t-deformation, and generalize Nakajima’s t-analogue of q-characters to thes...

Quantum cluster algebras