Let Gr be the affine Grassmannian for a connected complex reductive group G. Let CG be the complex vector space spanned by (equivalence classes of) Mirković-Vilonen cycles in Gr. The Beilinson-Drinfeld Grassmannian can be used to define a convolution product on MV-cycles, making CG into a commutative algebra. We show, in type A, that CG is isomorphic to C[N], the algebra of functions on the uni...

Given a finite dimensional algebra C (over an algebraically closed field) of global dimension at most two, we define its relation-extension algebra to be the trivial extension C ⋉ Ext 2 C (DC, C) of C by the CC -bimodule Ext 2 C (DC, C). We give a construction for the quiver of the relation-extension algebra in case the quiver of C has no oriented cycles. Our main result says that an algebrã C ...

In this paper we introduce noncommutative clusters and their mutations, which can be viewed as vast generalizations of both “classical” and quantum cluster structures. Each noncommutative cluster X is built on a torsion-free group G and a certain collection of its automorphisms. We assign to X a noncommutative algebra A(X) related to the group algebra of G, which is an analogue of the cluster a...

In this paper, we compute the Frobenius dimension of any cluster-tilted algebra finite type. Moreover, give conditions on bound quiver a [Formula: see text] such that has non-trivial open structures.

The purpose of this paper is to study the local structure semi-invariant picture a tame hereditary algebra near null root. Using construction that we call co-amalgamation, show completely described by pictures collection self-injective Nakayama algebras. We then describe cones using cluster-like structures support regular clusters. Finally, (piecewise linearly) invariant under cluster tilting.

We give an explicit description of all the exchange relations in any finite type cluster algebra with acyclic initial seed and principal coefficients.

In [2][3], the mathematicians Fomin and Zelevinsky described the mathematical object known as a quiver, and connected it with the theory of cluster algebras. In particular, each quiver can be represented by a seed of a cluster algebra, which couples a set of n variables with the adjacency matrix of the quiver. By performing a transformation on the quiver, we change accordingly the values of the...

We introduce signed exceptional sequences as factorizations of morphisms in the cluster morphism category. The objects of this category are wide subcategories of the module category of a hereditary algebra. A morphism [T ] : A → B is an equivalence class of rigid objects T in the cluster category of A so that B is the right hom-ext perpendicular category of the underlying object |T | ∈ A. Facto...

We compute the automorphism group of the affine surfaces with the coordinate ring isomorphic to a cluster algebra of rank 2.

The associahedron is an object that has been well studied and has numerous applications, particularly in the theory of operads, the study of non-crossing partitions, lattice theory and more recently in the study of cluster algebras. We approach the associahedron from the point of view of discrete homotopy theory. We study the abelianization of the discrete fundamental group, and show that it is...