相关论文: On cluster algebras from unpunctured surfaces with…
Kang, Kashiwara, Kim and Oh have proved that cluster monomials lie in the dual canonical basis, under a symmetric type assumption. This involves constructing a monoidal categorification of a quantum cluster algebra using representations of…
We generalize several recognizability theorems for free single-sorted algebras to the field of many-sorted algebras and provide, in a uniform way and without using neither regular tree grammars nor tree automata, purely algebraic proofs of…
Classification of cluster variables in cluster algebras (in particular, Grassmannian cluster algebras) is an important problem, which has direct application to computations of scattering amplitudes in physics. In this paper, we apply the…
We introduce admissible group actions on cluster algebras, cluster categories and quivers with potential and study the resulting orbit spaces. The orbit space of the cluster algebra has the structure of a generalized cluster algebra. This…
Among the mutation finite cluster algebras the tubular ones are a particularly interesting class. We show that all tubular (simply laced) cluster algebras are of exponential growth by two different methods: first by studying the…
A graded-division algebra is an algebra graded by a group such that all nonzero homogeneous elements are invertible. This includes division algebras equipped with an arbitrary group grading (including the trivial grading). We show that a…
The double point relation defines a natural theory of algebraic cobordism for bundles on varieties. We construct a simple basis (over the rationals) of the corresponding cobordism groups over Spec(C) for all dimensions of varieties and…
This is a survey paper of the theory of crystal bases, global bases and the cluster algebra structure on the quantum coordinate rings.
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 characterize vertex algebras (in a suitable sense) as algebras over a certain graded co-operad. We also discuss some examples and categorical implications of this characterization.
In this paper, we prove some combinatorial results on generalized cluster algebras. To be more precisely, we prove that (i) the seeds of a generalized cluster algebra $\mathcal A(\mathcal S)$ whose clusters contain particular cluster…
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…
We give the quantum Laurent expansion formula for the quantum cluster algebras from unpunctured orbifolds with arbitrary coefficients and quantization. As an application, positivity for such class of quantum cluster algebras is given. For…
Geiss-Leclerc-Schroer defined the cluster algebra structure on the coordinate ring $C[N(w)]$ of the unipotent subgroup, associated with a Weyl group element $w$ and they proved cluster monomials are contained in Lusztig's dual semicanonical…
The construction of partially compactified cluster algebras on coordinate rings is handled by using codimension 2 arguments on cluster covers. An analog of this in the quantum situation is highly desirable but has not been found yet. In…
It is a classical result that configuration spaces of labelled particles in $\mathbb{R}^d$ are free $E_d$-algebras and that their $d$-fold bar construction is equivalent to the $d$-fold suspension of the labelling space. In this paper, we…
We investigate the cluster-tilted algebras of finite representation type over an algebraically closed field. We give an explicit description of the relations for the quivers for finite representation type. As a consequence we show that a…
The classification, both up to isomorphism or up to equivalence, of the gradings on a finite dimensional nonassociative algebra A over an algebraically closed field F, such that its group scheme of automorphisms is smooth, is shown to be…
We construct a new class of symmetric algebras of tame representation type that are also the endomorphism algebras of cluster tilting objects in 2-Calabi-Yau triangulated categories, hence all their non-projective indecomposable modules are…
We study colored coverage and clustering problems. Here, we are given a colored point set where the points are covered by (unknown) $k$ clusters, which are monochromatic (i.e., all the points covered by the same cluster, have the same…