This is a concise expository survey of cluster algebras, introduced by S. Fomin and A. Zelevinsky in their four-part series of foundational papers [1], [2], [3], [4] (the paper [3] is with coauthor A. Berenstein). Our primary focus is on Cluster Algebras IV: Coefficients [4]. We introduce the setting of principal coefficients and define and study F-polynomials and g-vectors. Along the way, we m...

Abstract In earlier work, the author introduced a method for constructing Frobenius categorification of cluster algebra with frozen variables by starting from data an internally Calabi–Yau algebra, which becomes endomorphism cluster-tilting object in resulting category. this paper, we construct appropriate algebras polarized principal coefficients (which differ those addition more variables) an...

We prove that for a von Neumann algebra that is an algebraic K system with respect to some automorphism, the invariant state is K{clustering and r{clustering. Further we study in examples how far the von Neumann algebra inherits the K property from the underlying C algebra.

Let A be a finite dimensional hereditary algebra over an algebraically closed field k, A(m) be the m-replicated algebra of A and Cm(A) be the m-cluster category of A. We investigate properties of complements to a faithful almost complete tilting A(m)-module and prove that the m-cluster mutation in Cm(A) can be realized in mod A (m), which generalizes corresponding results on duplicated algebras...

Lots of research focuses on the combinatorics behind various bases of cluster algebras. This paper studies the natural basis of a type A cluster algebra, which consists of all cluster monomials. We introduce a new kind of combinatorial formula for the cluster monomials in terms of the so-called globally compatible collections. We give bijective proofs of these formulas by comparing with the wel...

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...

We give the path model solution for the cluster algebra variables of the T system of type Ar with generic boundary conditions. The solutions are partition functions of (strongly) non-intersecting paths on weighted graphs. The graphs are the same as those constructed for the Q-system in our earlier work, and depend on the seed or initial data in terms of which the solutions are given. The weight...