Related papers: Preprojective algebras and cluster algebras
We give an introduction to the theory of cluster categories and cluster tilted algebras. We include some background on the theory of cluster algebras, and discuss the interplay with cluster categories and cluster tilted algebras.
We formulate a relative, representation theoretic, notion of the algebraic cone construction. This motivates a generalization of the cone corresponding to a preprojective algebra.
We start with definitions of the general notions of the theory of $\Bbb Z_{2}$-graded algebras. Then we consider theory of inductive families of $\Bbb Z_{2}$-graded semisimple finite-dimensional algebras and its representations in the…
The survey contains a brief description of the ideas, constructions, results, and prospects of the theory of hypergroups and generalized translation operators. Representations of hypergroups are considered, being treated as continuous…
We develop a structure theory of connected solvable spherical subgroups in semisimple algebraic groups. Based on this theory, we obtain an explicit classification of all such subgroups up to conjugation.
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…
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…
In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.
We generalize to the non simply-laced case results of Gei\ss, Leclerc and Schr\"oer about the cluster structure of the coordinate ring of the maximal unipotent subgroups of simple Lie groups. In this way, cluster structures in the non…
This is a first step guide to the theory of cluster algebras. We especially focus on basic notions, techniques, and results concerning seeds, cluster patterns, and cluster algebras.
We review how the (quantum) cluster algebra naturally emerges in the study of four-dimensional $\mathcal{N}=2$ supersymmetric gauge theories.
Cluster-tilted algebras are trivial extensions of tilted algebras. This correspondence induces a surjective map from tilted algebras to cluster-tilted algebras. If B is a cluster-tilted algebra, we use the fibre of B under this map to study…
In this paper we apply a recently proposed algebraic theory of integration to projective group algebras. These structures have received some attention in connection with the compactification of the $M$ theory on noncommutative tori. This…
We construct the intermediate coverings of cluster-tilted algebras by defining the generalized cluster categories. These generalized cluster categories are Calabi-Yau triangulated categories with fraction CY-dimension and have also cluster…
Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type ADE can be recovered from the data of the corresponding…
In the theory of generalized cluster algebras, we build the so-called cluster formula and $D$-matrix pattern. Then as applications, some fundamental conjectures of generalized cluster algebras are solved affirmatively.
A series of nonrepresentable relation algebras is constructed from groups. We use them to prove that there are continuum many subvarieties between the variety of representable relation algebras and the variety of coset relation algebras. We…
We introduce the notion of a lower bound cluster algebra generated by projective cluster variables as a polynomial ring over the initial cluster variables and the so-called projective cluster variables. We show that under an acyclicity…
New notions are introduced in algebra in order to better study the congruences in number theory. For example, the <special semigroups> makes an important such contribution.
We survey some recent constructions of cluster algebra structures on coordinate rings of unipotent subgroups and unipotent cells of Kac-Moody groups. We also review a quantized version of these results.