Related papers: From iterated tilted algebras to cluster-tilted al…
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.
Higher Auslander algebras were introduced by Iyama generalizing classical concepts from representation theory of finite dimensional algebras. Recently these higher analogues of classical representation theory have been increasingly studied.…
This paper develops the theory of KLR algebras with a Dynkin diagram automorphism. This is foundational material intended to allow folding techniques in the theory of KLR algebras.
Some quantum algebras build from deformed oscillator algebras may be described in terms of a particular case of extended umbral calculus. We give here an example of a specific relation between such certain quantum algebras and generalized…
We complete classification of mutation-finite cluster algebras by extending the technique derived by Fomin, Shapiro, and Thurston to skew-symmetrizable case. We show that for every mutation-finite skew-symmetrizable matrix a diagram…
Let D be the cluster category of Dynkin type A_{\infty}. This paper provides a bijection between torsion theories in D and certain configurations of arcs connecting non-neighbouring integers.
In this paper, we introduce an algebra structure denoted by InvDer algebra whose which we twist an algebra thanks to an invertible derivation, where its inverse is also a derivation. We define InvDer Lie algebras, InvDer associated…
This is a survey article on some connections between cluster algebras and link invariants, written for the Notices of the AMS.
We introduce a class of non-commutative algebras that carry a non-commutative (geometric) cluster structure which are generated by identical copies of generalized Weyl algebras. Equivalent conditions for the finiteness of the set of the…
This is a survey on recent developments in Cohen-Macaulay representations via tilting and cluster tilting theory. We explain triangle equivalences between the singularity categories of Gorenstein rings and the derived (or cluster)…
We consider finite iterated generalized harmonic sums weighted by the binomial $\binom{2k}{k}$ in numerators and denominators. A large class of these functions emerges in the calculation of massive Feynman diagrams with local operator…
We propose a generic framework to obtain certain types of contracted and centrally extended algebras. This is based on the existence of quadratic algebras (reflection algebras and twisted Yangians), naturally arising in the context of…
An involutive Lie bialgebra induces a Batalin-Vilkovisky operator on its exterior algebra. We introduce a graded generalization of the necklace Lie bialgebra, which depends on a choice of a quiver $Q$. We relate the resulting…
The paper explores categorical interconnections between lattice-valued Relational systems and algebras of Fitting's lattice-valued modal logic. We define lattice-valued boolean systems, and then we study co-adjointness, adjointness of…
We show that $\tau$-tilting finite simply connected algebras are representation-finite. Then, some related algebras are considered, including iterated tilted algebras, tubular algebras and so on. We also prove that the $\tau$-tilting…
We give a complete description of the cluster-mutation classes of diagrams of Dynkin types \mathbb{A},\mathbb{B},\mathbb{D} and of affine Dynkin types \mathbb{B}^{(1)},\mathbb{C}^{(1)},\mathbb{D}^{(1)} via certain families of diagrams.
In this paper, we study two-sided tilting complexes of preprojective algebras of Dynkin type. We construct the most fundamental class of two-sided tilting complexes, which has a group structure by derived tensor products and induces a group…
In present work, we find a class of Lie algebras, which are defined from the symmetrizable generalized intersection matrices. However, such algebras are different from generalized intersection matrix algebras and intersection matrix…
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…
In the present paper we examine the relationship between several type $A$ cluster theories and structures. We define a 2D geometric model of a cluster theory, which generalizes cluster algebras from surfaces, and encode several existing…