Let V be a k-dimensional complex vector space. The Plücker ring of polynomial SL(V ) invariants of a collection of n vectors in V can be alternatively described as the homogeneous coordinate ring of the Grassmannian Gr(k, n). In 2003, using combinatorial tools developed by A. Postnikov, J. Scott showed that the Plücker ring carries a cluster algebra structure. Over the ensuing decade, this has ...

We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebra does not depend on the choice of coefficients, describe this complex explicitly in terms of “tagged triangulations” of the surface, and determine its homotopy type and its growth rate.

Large-scale RDF graph databases stored in shared-nothing clusters require query processing engine that can effectively exploit highly parallel computation environment. We propose algebra of RDF graphs and its physical counterpart, physical algebra of RDF graphs, designed to implement queries as distributed dataflow programs that run on cluster of servers. Operations of algebra reflect the chara...

We study monoidal categorifications of certain subcategories C J $\mathcal {C}_J$ finite-dimensional modules over quantum affine algebras, whose cluster algebra structures on their Grothendieck rings K ( ) $K(\mathcal {C}_J)$ are closely related to the category quiver Hecke type A ∞ $A_\infty$ via generalized Schur–Weyl duality functors. In particular, when is $A$ or B $B$ , subcategory coincid...

We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebra does not depend on the choice of coefficients, describe this complex explicitly in terms of “tagged triangulations” of the surface, and determine its homotopy type and its growth rate.

We give an example of a finite-dimensional algebra with 2-cluster tilting module and simple which has infinite complexity. This answers question Erdmann Holm.

We prove a multiplication theorem for quantum cluster algebras of acyclic quivers. The theorem generalizes the multiplication formula for quantum cluster variables in [19]. Moreover some ZP-bases in quantum cluster algebras of finite and affine types are constructed. Under the specialization q and coefficients to 1, these bases are the integral bases of cluster algebra of finite and affine type...