Quantum cluster algebras

Cluster algebras form an axiomatically defined class of commutative rings designed to serve as an algebraic framework for the theory of total positivity and canonical bases in semisimple groups and their quantum analogs. In this paper we introduce and study quantum deformations of cluster algebras.     

Connected quantized Weyl algebras and quantum cluster algebras

For an algebraically closed field $\mathbb{K}$, we investigate a class of noncommutative $\mathbb{K}$-algebras called connected quantized Weyl algebras. Such an algebra has a PBW basis for a set of generators $\{x_1,\dots ,x_n\}$ such that each pair satisfies a relation of the form $x_i{x}_j=q_{ij}{x}_j{x}_i+r_{ij}$, where $q_{ij}\in {\mathbb{K}}^{?}$ and $r_{ij}\in \mathbb{K}$, with, in some sense, sufficiently many pairs for which $r_{ij}\ne 0$. For such an algebra it turns out that there is a single parameter q such that each $q_{ij}=q^{\pm 1}$. Assuming that $q\ne \pm 1$, we classify connected quantized Weyl algebras, showing that there are two types linear and cyclic. When q is not a root of unity we determine the prime spectra for each type. The linear case is the easier, although the result depends on the parity of n, and all prime ideals are completely prime. In the cyclic case, which can only occur if n is odd, there are prime ideals for which the factors have arbitrarily large Goldie rank.We apply connected quantized Weyl algebras to obtain presentations of two classes of quantum cluster algebras. Let $n\ge 3$ be an odd integer. We present the quantum cluster algebra of a Dynkin quiver of type $A_{n?1}$ as a factor of a linear connected quantized Weyl algebra by an ideal generated by a central element. We also consider the quiver $P_{n+1}^{(1)}$ identified by Fordy and Marsh in their analysis of periodic quiver mutation. When n is odd, we show that the quantum cluster algebra of this quiver is generated by a cyclic connected quantized Weyl algebra in n variables and one further generator. We also present it as the factor of an iterated skew polynomial algebra in $n+2$ variables by an ideal generated by a central element. For both classes, the quantum cluster algebras are simple noetherian.We establish Poisson analogues of the results on prime ideals and quantum cluster algebras. We determine the Poisson prime spectra for the semiclassical limits of the linear and cyclic connected quantized Weyl algebras and show that, when n is odd, the cluster algebras of $A_{n?1}$ and $P_{n+1}^{(1)}$ are simple Poisson algebras that can each be presented as a Poisson factor of a polynomial algebra, with an appropriate Poisson bracket, by a principal ideal generated by a Poisson central element.     

On quantum mechanics and generalized Clifford algebras

In this note we give elementary examples of the naturalness of generalized Clifford algebras appearance, in some particular quantum mechanical models. First Weyl’s program [1] for quantum kinematics for the case of simplest Galois fieldsZ _n is realized in terms of generalized Clifford algebras. Dynamics might then be introduced, following the ideas of Hanney and Berry [2], as shown in [3]. Second the coherent state picture of the finite dimensional “Z _n — Quantum Mechanics” is presented. In the last part the known coherent states ofq-deformed quantum oscillators (q≡ω) are explicitly shown in the generalized Grassman algebras and the generalized Clifford algebras settings. Presented atThe Polish-Mexican Seminar, Kazimierz Dolny, August 1998 — Poland. 176     

On quantum cluster algebras of finite type

We extend the definition of a quantum analogue of the Caldero-Chapoton map defined by D. Rupel. When Q is a quiver of finite type, we prove that the algebra (Q) generated by all cluster characters is exactly the quantum cluster algebra (Q).     

Fixed rings of quantum generalized Weyl algebras

Abstract

Generalized Weyl algebras (GWAs) appear in diverse areas of mathematics including mathematical physics, noncommutative algebra, and representation theory. We study the invariants of quantum GWAs under finite order automorphisms. We extend a theorem of Jordan and Wells and apply it to determine the fixed ring of quantum GWAs under diagonal automorphisms. We further study properties of the fixed rings including global dimension, the Calabi–Yau property, rigidity, and simplicity.     

Shellability and higher Cohen-Macaulay connectivity of generalized cluster complexes

Let Φ be a finite root system of rank n and let m be a nonnegative integer. The generalized cluster complex Δ^m(Φ) was introduced by S. Fomin and N. Reading. It was conjectured by these authors that Δ^m(Φ) is shellable and by V. Reiner that it is (m + 1)-Cohen-Macaulay, in the sense of Baclawski. These statements are proved in this paper. Analogous statements are shown to hold for the positive part Δ₊^m(Φ) of Δ^m(Φ). An explicit homotopy equivalence is given between Δ₊^m(Φ) and the poset of generalized noncrossing partitions, associated to the pair (Φ, m) by D. Armstrong.     

Graded quantum cluster algebras of infinite rank as colimits

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 cluster algebras of finite rank.As an application, for each k we construct a graded quantum infinite Grassmannian admitting a cluster algebra structure, extending an earlier construction of the authors for $k=2$.     

Cluster tilting objects in generalized higher cluster categories

We prove the existence of an m-cluster tilting object in a generalized m-cluster category which is (m+1)-Calabi-Yau and Hom-finite, arising from an (m+2)-Calabi-Yau dg algebra. This is a generalization of the result for the m=1 case in Amiot’s Ph.D. thesis. Our results apply in particular to higher cluster categories associated to Ginzburg dg categories coming from suitable graded quivers with superpotential, and higher cluster categories associated to suitable finite-dimensional algebras of finite global dimension.     

Examples of quantum cluster algebras associated to partial flag varieties

We give several explicit examples of quantum cluster algebra structures, as introduced by Berenstein and Zelevinsky, on quantized coordinate rings of partial flag varieties and their associated unipotent radicals. These structures are shown to be quantizations of the cluster algebra structures found on the corresponding classical objects by Geiß, Leclerc and Schröer, whose work generalizes that of several other authors. We also exhibit quantum cluster algebra structures on the quantized enveloping algebras of the Lie algebras of the unipotent radicals.     

A quantum analogue of generic bases for affine cluster algebras

We construct quantized versions of generic bases in quantum cluster algebras of finite and affine types.Under the specialization of q and coefficients to 1,these bases are generic bases of finite and affine cluster algebras.     

Graded quiver varieties,quantum cluster algebras and dual canonical basis

Inspired by a previous work of Nakajima, we consider perverse sheaves over acyclic graded quiver varieties and study the Fourier–Sato–Deligne transform from a representation theoretic point of view. We obtain deformed monoidal categorifications of acyclic quantum cluster algebras with specific coefficients. In particular, the (quantum) positivity conjecture is verified whenever there is an acyclic seed in the (quantum) cluster algebra.     

Geometric construction of crystal bases for quantum generalized Kac-Moody algebras

We provide a geometric realization of the crystal B(∞) for quantum generalized Kac-Moody algebras in terms of the irreducible components of certain Lagrangian subvarieties in the representation spaces of a quiver.     

Canonical bases for quantum generalized Kac–Moody algebras

We construct canonical bases for quantum generalized Kac–Moody algebras using semisimple perverse sheaves.     

Power algebras and generalized quotient algebras

The notion of a good quotient relation has been introduced as an attempt to generalize the notion of a quotient algebra to relations on an algebra which are not necessarily congruences. In order to make it possible to prove generalized versions of 'power isomorphism theorems', the more restrictive notions of very good, Hoare good and Smyth good relation have been introduced. In this paper we describe the relationships between Hoare good, Smyth good and very good relations. As a consequence, we prove that every structure preserving relation on an algebra is very good. Received September 25, 1998; accepted in final form January 14, 1999.     

Auslander algebras and initial seeds for cluster algebras

Let Q be a Dynkin quiver and the corresponding set of positiveroots. For the preprojective algebra associated to Q, a rigid-module I_Q is produced with r = || pairwise non-isomorphic indecomposabledirect summands by pushing the injective modules of the Auslanderalgebra of k Q to . If N is a maximal unipotent subgroup ofa complex simply connected simple Lie group of type |Q|, thenthe coordinate ring [N] is an upper cluster algebra. It is shownthat the elements of the dual semicanonical basis which correspondto the indecomposable direct summands of I_Q coincide with certaingeneralized minors which form an initial cluster for [N] andthat the corresponding exchange matrix of this cluster can beread from the Gabriel quiver of End(I_Q). Finally, the fact thatthe categories of injective modules over and over its covering are triangulated is exploited in order to show several interesting identities in the respectivestable module categories.     

Quantum algebras,quantum coalgebras,invariants of 1-1 tangles and knots

Quantum coalgebras are defined and studied. A theory of asso­ciated invariants of 1-1 tangles, knots and links is developed. The notion of quantum coalgebra is more general than dual of quantum algebra. Examples of quantum algebras include quasitriangular Hopf algebras and examples of quantum coalgebras include coquasi triangu­lar Hopf algebras.