相关论文: Root of unity quantum cluster algebras and Cayley-…
We describe a connection between the subjects of cluster algebras, polynomial identity algebras and discriminants. For this, we define the notion of root of unity quantum cluster algebras and prove that they are polynomial identity…
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…
There are two main types of objects in the theory of cluster algebras: the upper cluster algebras ${{\boldsymbol{\mathsf U}}}$ with their Gekhtman-Shapiro-Vainshtein Poisson brackets and their root of unity quantizations…
All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are proved to possess quantum cluster algebra structures under mild conditions. Furthermore, it is shown that these quantum cluster algebras always…
Our motivation is to build a systematic method in order to investigate the structure of cluster algebras of geometric type. The method is given through the notion of mixing-type sub-seeds, the theory of seed homomorphisms and the view-point…
It is known that many (upper) cluster algebras possess different kinds of good bases which contain the cluster monomials and are parametrized by the tropical points of cluster Poisson varieties. For a large class of upper cluster algebras…
In this article, both versions of multiparameter quantum Weyl algebras have been studied at the roots of unity. The center, PI degree, maximal-dimensional simple modules, and Azumaya locus have been explicitly computed for such algebras.
We construct common triangular bases for almost all the known (quantum) cluster algebras from Lie theory. These bases provide analogs of the dual canonical bases, long anticipated in cluster theory. In cases where the generalized Cartan…
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 formalize the way in which one can think about cluster algebras of infinite rank by showing that every rooted cluster algebra of infinite rank can be written as a colimit of rooted cluster algebras of finite rank. Relying on the proof of…
We provide a graded and quantum version of the category of rooted cluster algebras introduced by Assem, Dupont and Schiffler and show that every graded quantum cluster algebra of infinite rank can be written as a colimit of graded quantum…
We introduce a framework for $\mathbb{Z}$-gradings on cluster algebras (and their quantum analogues) that are compatible with mutation. To do this, one chooses the degrees of the (quantum) cluster variables in an initial seed subject to a…
We compute the center and Azumaya locus in the simplest non-abelian examples of quantized multiplicative quiver varieties at a root of unity: quantum Weyl algebras of rank $N$, and quantum differential operators on the quantum group…
A family of quantum cluster algebras is introduced and studied. In general, these algebras are new, but subclasses have been studied previously by other authors. The algebras are indexed by double partitions or double flag varieties.…
We develop a Poisson geometric framework for studying the representation theory of all contragredient quantum super groups at roots of unity. This is done in a uniform fashion by treating the larger class of quantum doubles of bozonizations…
We prove that the quantum cluster algebra structure of a unipotent quantum coordinate ring $A_q(\mathfrak{n}(w))$, associated with a symmetric Kac-Moody algebra and its Weyl group element $w$, admits a monoidal categorification via the…
We use stratified Morse theory to construct a complex to compute the cohomology of the complement of a hyperplane arrangement with coefficients in a complex rank one local system. The linearization of this complex is shown to be the…
This paper is devoted to finding solutions of polynomial equations in roots of unity. It was conjectured by S. Lang and proved by M. Laurent that all such solutions can be described in terms of a finite number of parametric families called…
We describe the center of quantum tori appearing in quantum higher Teichm\"uller theory when the quantum parameter is an even root of unity. This is a subsequent of the work in the odd roots of unity case. The centers of the quantum tori…
The question of embedding fields into central simple algebras $B$ over a number field $K$ was the realm of class field theory. The subject of embedding orders contained in the ring of integers of maximal subfields $L$ of such an algebra…