Quantum Grothendieck rings as 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...

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

Cluster algebras and quantum affine algebras

Let C be the category of finite-dimensional representations of a quantum affine algebra Uq(ĝ) of simply-laced type. We introduce certain monoidal subcategories Cl (l ∈ N) of C and we study their Grothendieck rings using cluster algebras.

Double-partition Quantum Cluster Algebras

A family of quantum cluster algebras is introduced and studied. In general, these algebras are new, but sub-classes have been studied previously by other authors. The algebras are indexed by double partitions or double flag varieties. Equivalently, they are indexed by broken lines L. By grouping together neighboring mutations into quantum line mutations we can mutate from the cluster algebra of...