Related papers: A=U for Locally Acyclic Cluster Algebras
We give characterizations of unital uniform topological algebras and saturated locally multiplicatively convex algebras by means of multiplicative linear functionals. Some automatic continuity theorems in advertibly complete uniform…
We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation…
The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra A of finite type can be realized as a Hall algebra, called the exceptional Hall algebra, of…
Cluster ensemble is a pair of positive spaces (X, A) related by a map p: A -> X. It generalizes cluster algebras of Fomin and Zelevinsky, which are related to the A-space. We develope general properties of cluster ensembles, including its…
In the present paper the notion of a Hilbert module over a locally C*-algebra is discussed and some results are obtained on this matter. In particular, we give a detailed proof of the known result that the set of adjointable endomorphisms…
We prove that the set of c-vectors of the cluster algebra associated to an acyclic quiver Q coincides with the set of real Schur roots and their opposites in the root system associated to Q.
We find and discuss an unexpected (to us) order $n$ cyclic group of automorphisms of the Lie algebra $I\mathfrak{u}_n := \mathfrak{u}_n\ltimes\mathfrak{u}_n^\ast$, where $\mathfrak{u}_n$ is the Lie algebra of upper triangular $n\times n$…
We clarify the natural cluster algebra of type A that exists in a residual and tropical form in the kinematical space as suggested in 1711.09102 by the use of triangulations, mutations and associahedron on the definition of scattering…
Cluster algebras are a class of commutative algebras whose generators are defined by a recursive process called mutation. We give a brief introduction to cluster algebras, and explain how discrete integrable systems can appear in the…
We give a geometric interpretation of cluster varieties in terms of blowups of toric varieties. This enables us to provide, among other results, an elementary geometric proof of the Laurent phenomenon for cluster algebras (of geometric…
It is proved that the K_0-group of a cluster C*-algebra is isomorphic to the corresponding cluster algebra. As a corollary, one gets a shorter proof of the positivity conjecture for cluster algebras. As an example, we consider a cluster…
We initiate a systematic study of the cohomology of cluster varieties. We introduce the Louise property for cluster algebras that holds for all acyclic cluster algebras, and for most cluster algebras arising from marked surfaces. For…
We give a brief introduction to (upper) cluster algebras and their quantization using examples. Then we present several important families of bases for these algebras using topological models. We also discuss tropical properties of these…
Let $Q$ be a finite acyclic valued quiver. We give the cluster multiplication formulas in the quantum cluster algebra of $Q$ with arbitrary coefficients, by applying certain quotients of derived Hall subalgebras of $Q$. These formulas can…
We compute the automorphism group of the affine surfaces with the coordinate ring isomorphic to a cluster algebra of rank 2.
Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert…
We show that the inclusion of an affine Hecke algebra in its Schwartz completion induces an isomorphism on periodic cyclic homology.
We study the cluster automorphism group of a skew-symmetric cluster algebra with geometric coefficients. For this, we introduce the notion of gluing free cluster algebra, and show that under a weak condition the cluster automorphism group…
We establish a bijective correspondence between certain non-self-intersecting curves in an $n$-punctured disc and positive ${\mathbf c}$-vectors of acyclic cluster algebras whose quivers have multiple arrows between every pair of vertices.…
A few properties of unitary Cayley graphs are explored using their eigenvalues. It is shown that the adjacency algebra of a unitary Cayley graph is a coherent algebra. Finally, a class of unitary Cayley graphs that are distance regular are…