Motivated by a recent conjecture by Hernandez and Leclerc [arXiv:0903.1452], we embed a Fomin-Zelevinsky cluster algebra [arXiv:math/0104151] into the Grothendieck ring R of the category of representations of quantum loop algebras U_q(Lg) of a symmetric Kac-Moody Lie algebra g, studied earlier by the author via perverse sheaves on graded quiver varieties [arXiv:math/9912158]. Graded quiver varieties controlling the image can be identified with varieties which Lusztig used to define the canonical base. The cluster monomials form a subset of the base given by the classes of simple modules in R, or Lusztig's dual canonical base. The positivity and linearly independence (and probably many other) conjectures of cluster monomials [arXiv:math/0104151] follow as consequences, when there is a seed with a bipartite quiver. Simple modules corresponding to cluster monomials factorize into tensor products of `prime' simple ones according to the cluster expansion.

Quiver varieties and cluster algebras

Motivated by a recent conjecture by Hernandez and Leclerc, we embed a Fomin-Zelevinsky cluster algebra into the Grothendieck ring R of the category of representations of quantum loop algebras Uq(Lg) of a symmetric Kac-Moody Lie algebra, studied earlier by the author via perverse sheaves on graded quiver varieties. Graded quiver varieties controlling the image can be identified with varieties which Lusztig used to define the canonical base. The cluster monomials form a subset of the base given by the classes of simple modules in R, or Lusztig’s dual canonical base. The conjectures that cluster monomials are positive and linearly independent (and probably many other conjectures) of Fomin and Zelevinsky follow as consequences when there is a seed with a bipartite quiver. Simple modules corresponding to cluster monomials factorize into tensor products of “prime” simple ones according to the cluster expansion.

/pdf/quiver-varieties-and-cluster-algebras-1w2t3sivuv.pdf

/pdf/monoidal-categorification-of-cluster-algebras-10nu1c8v5e.pdf

Monoidal categorification of cluster algebras

These are notes for a series of lectures presented at the ASIDE conference 2016. The definition of a cluster algebra is motivated through several examples, namely Markov triples, the Grassmannians $\mathit{Gr}_{2}(\mathbb{C}^{n})$, and the appearance of double Bruhat cells in the theory of total positivity. Once the definition of cluster algebras is introduced in several stages of increasing generality, proofs of fundamental results are sketched in the rank 2 case. From these foundations we build up the notion of Poisson structures compatible with a cluster algebra structure and indicate how this leads to a quantization of cluster algebras. Finally we give applications of these ideas to integrable systems in the form of Zamolodchikov periodicity and the pentagram map.

Introduction to Cluster Algebras

We consider the quantum cluster algebras which are injective-reachable and introduce a triangular basis in every seed. We prove that, under some initial conditions, there exists a unique common triangular basis with respect to all seeds. This basis is parametrized by tropical points as expected in the Fock-Goncharov conjecture. 
As an application, we prove the existence of the common triangular bases for the quantum cluster algebras arising from representations of quantum affine algebras and partially for those arising from quantum unipotent subgroups. This result implies monoidal categorification conjectures of Hernandez-Leclerc and Fomin-Zelevinsky in the corresponding cases: all cluster monomials correspond to simple modules.

Triangular bases in quantum cluster algebras and monoidal categorification conjectures

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. As byproducts, we obtain the existence of counting polynomials for these varieties and the positivity conjecture with respect to acyclic seeds. These results complete previous work by Caldero and Reineke and confirm a recent conjecture by Rupel.

Quantum Cluster Variables via Serre Polynomials

We consider the quantum cluster algebras which are injective-reachable and introduce a triangular basis in every seed. We prove that, under some initial conditions, there exists a unique common triangular basis with respect to all seeds. This basis is parameterized by tropical points as expected in the Fock–Goncharov conjecture. As an application, we prove the existence of the common triangular bases for the quantum cluster algebras arising from representations of quantum affine algebras and partially for those arising from quantum unipotent subgroups. This result implies monoidal categorification conjectures of Hernandez and Leclerc and Fomin and Zelevinsky in the corresponding cases: all cluster monomials correspond to simple modules.

It is known that many (upper) cluster algebras possess different kinds of good bases which contain the cluster monomials and are parametrized by the tropical points of cluster Poisson varieties. For a large class of upper cluster algebras (injective-reachable ones with full rank coefficients), we describe all of its bases with these properties. Moreover, we show the existence of the generic basis for them. In addition, we prove that Bridgeland's representation theoretic formula is effective for their theta functions (weak genteelness). 
Our results apply to (almost) all well-known cluster algebras arising from representation theory or higher Teichmuller theory, including quantum affine algebras, unipotent cells, double Bruhat cells, skein algebras over surfaces, where we change the coefficients if necessary so that the full rank assumption holds.

Bases for upper cluster algebras and tropical points

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 generalize the result of that paper to any acyclic quantum cluster algebra with arbitrary nondegenerate coefficients. In particular, we obtain the generic basis, the dual PBW basis, and the dual canonical basis. The method consists in a correction technique, which works for general quantum cluster algebras.

$t$-Analog of $q$-Characters, Bases of Quantum Cluster Algebras, and a Correction Technique

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Papers

Triangular bases in quantum cluster algebras and monoidal categorification conjectures

Quantum Cluster Variables via Serre Polynomials

Triangular bases in quantum cluster algebras and monoidal categorification conjectures

Bases for upper cluster algebras and tropical points

$t$-Analog of $q$-Characters, Bases of Quantum Cluster Algebras, and a Correction Technique