scholarly journals Generalised cluster algebras and $q$-characters at roots of unity

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Anne-Sophie Gleitz
Keyword(s):  
Cluster Algebra ◽  
Integral Form ◽  
The Other ◽  
Cluster Algebras ◽  
Roots Of Unity ◽  
Quantum Affine Algebra ◽  
Affine Algebra ◽  
Grothendieck Ring ◽  
Root Of Unity ◽  
International Audience

International audience Shapiro and Chekhov (2011) have introduced the notion of <i>generalised cluster algebra</i>; we focus on an example in type $C_n$. On the other hand, Chari and Pressley (1997), as well as Frenkel and Mukhin (2002), have studied the <i>restricted integral form</i> $U^{\mathtt{res}}_ε (\widehat{\mathfrak{g}})$ of a quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ where $q=ε$ is a root of unity. Our main result states that the Grothendieck ring of a tensor subcategory $C_{ε^\mathbb{z}}$ of representations of $U^{\mathtt{res}}_ε (L\mathfrak{sl}_2)$ is a generalised cluster algebra of type $C_{l−1}$, where $l$ is the order of $ε^2$. We also state a conjecture for $U^{\mathtt{res}}_ε (L\mathfrak{sl}_3)$, and sketch a proof for $l=2$. Shapiro et Chekhov (2011) ont introduit la notion d'<i>algèbre amassée généralisée</i>; nous étudions un exemple en type $C_n$. Par ailleurs, Chari et Pressley (1997), ainsi que Frenkel et Mukhin (2002), ont étudié la <i>forme entière restreinte</i> $U^{\mathtt{res}}_ε (\widehat{\mathfrak{g}})$ d'une algèbre affine quantique $U_q(\widehat{\mathfrak{g}})$ où $q=ε$ est une racine de l'unité. Notre résultat principal affirme que l'anneau de Grothendieck d'une sous-catégorie tensorielle $C_{ε^\mathbb{z}}$ de représentations de $U^{\mathtt{res}}_ε (L\mathfrak{sl}_2)$ est une algèbre amassée généralisée de type $C_{l−1}$, où $l$ est l'ordre de $ε^2$. Nous conjecturons une propriété similaire pour $U^{\mathtt{res}}_ε (L\mathfrak{sl}_3)$ et donnons un aperçu de la preuve pour $l=2$.

scholarly journals Periodicities of T-systems and Y-systems

2010 ◽  
Vol 197 ◽  
pp. 59-174 ◽  
Author(s):  
Rei Inoue ◽  
Osamu Iyama ◽  
Atsuo Kuniba ◽  
Tomoki Nakanishi ◽  
Junji Suzuki
Keyword(s):  
Lie Algebra ◽  
Direct Method ◽  
Cluster Algebra ◽  
Quantum Affine Algebra ◽  
Cluster Category ◽  
Affine Algebra ◽  
Grothendieck Ring ◽  
Finite Dimensional ◽  
Level 2 ◽  
T System

The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of Yangian or quantum affine algebra associated with a complex simple Lie algebra. The unrestricted T-system admits a reduction called the restricted T-system. In this paper we formulate the periodicity conjecture for the restricted T-systems, which is the counterpart of the known and partially proved periodicity conjecture for the restricted Y-systems. Then, we partially prove the conjecture by various methods: the cluster algebra and cluster category method for the simply laced case, the determinant method for types A and C, and the direct method for types A, D, and B (level 2).

scholarly journals A quantum cluster algebra approach to representations of simply laced quantum affine algebras

2020 ◽  
Author(s):  
Léa Bittmann
Keyword(s):  
Cluster Algebra ◽  
Irreducible Representations ◽  
Algebra Structure ◽  
Quantum Affine Algebra ◽  
Affine Algebra ◽  
Grothendieck Ring ◽  
Finite Dimensional ◽  
Quantum Cluster Algebra ◽  
Algebra Approach ◽  
Quantum Cluster

AbstractWe establish a quantum cluster algebra structure on the quantum Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional representations of a simply-laced quantum affine algebra. Moreover, the (q, t)-characters of certain irreducible representations, among which fundamental representations, are obtained as quantum cluster variables. This approach gives a new algorithm to compute these (q, t)-characters. As an application, we prove that the quantum Grothendieck ring of a larger category of representations of the Borel subalgebra of the quantum affine algebra, defined in a previous work as a quantum cluster algebra, contains indeed the well-known quantum Grothendieck ring of the category of finite-dimensional representations. Finally, we display our algorithm on a concrete example.

scholarly journals Periodicities of T-systems and Y-systems

2010 ◽  
Vol 197 ◽  
pp. 59-174 ◽  
Author(s):  
Rei Inoue ◽  
Osamu Iyama ◽  
Atsuo Kuniba ◽  
Tomoki Nakanishi ◽  
Junji Suzuki
Keyword(s):  
Lie Algebra ◽  
Direct Method ◽  
Cluster Algebra ◽  
Quantum Affine Algebra ◽  
Cluster Category ◽  
Affine Algebra ◽  
Grothendieck Ring ◽  
Finite Dimensional ◽  
Level 2 ◽  
T System

The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of Yangian or quantum affine algebra associated with a complex simple Lie algebra. The unrestricted T-system admits a reduction called the restricted T-system. In this paper we formulate the periodicity conjecture for the restricted T-systems, which is the counterpart of the known and partially proved periodicity conjecture for the restricted Y-systems. Then, we partially prove the conjecture by various methods: the cluster algebra and cluster category method for the simply laced case, the determinant method for typesAandC, and the direct method for typesA, D, andB(level 2).

scholarly journals CLUSTER AUTOMORPHISMS AND COMPATIBILITY OF CLUSTER VARIABLES

2014 ◽  
Vol 56 (3) ◽  
pp. 705-720 ◽  
Author(s):  
IBRAHIM ASSEM ◽  
VASILISA SHRAMCHENKO ◽  
RALF SCHIFFLER
Keyword(s):  
Cluster Algebra ◽  
The Other ◽  
Cluster Algebras ◽  
Dynkin Type ◽  
Rank 2

AbstractIn this paper, we introduce a notion of unistructural cluster algebras, for which the set of cluster variables uniquely determines the clusters, as well as the notion of weak unistructural cluster algebras, for which the set of cluster variables determines the clusters, provided that the type of the cluster algebra is fixed. We prove that, for cluster algebras of the Dynkin type, the two notions of unistructural and weakly unistructural coincide, and that cluster algebras of rank 2 are always unistructural. We then prove that a cluster algebra $\mathcal A$ is weakly unistructural if and only if any automorphism of the ambient field, which restricts to a permutation of cluster variables of $\mathcal A$, is a cluster automorphism. We also investigate the Fomin-Zelevinsky conjecture that two cluster variables are compatible if and only if one does not appear in the denominator of the Laurent expansions of the other.

scholarly journals The cluster and dual canonical bases of Z [x_11, ..., x_33] are equal

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Brendon Rhoades
Keyword(s):  
Quantum Group ◽  
Polynomial Ring ◽  
Geometric Model ◽  
Cluster Algebra ◽  
Canonical Basis ◽  
The Other ◽  
Monomial Basis ◽  
Dual Canonical Basis ◽  
Nous Montrons ◽  
International Audience

International audience The polynomial ring $\mathbb{Z}[x_{11}, . . . , x_{33}]$ has a basis called the dual canonical basis whose quantization facilitates the study of representations of the quantum group $U_q(\mathfrak{sl}3(\mathbb{C}))$. On the other hand, $\mathbb{Z}[x_{11}, . . . , x_{33}]$ inherits a basis from the cluster monomial basis of a geometric model of the type $D_4$ cluster algebra. We prove that these two bases are equal. This extends work of Skandera and proves a conjecture of Fomin and Zelevinsky. This also provides an explicit factorization of the dual canonical basis elements of $\mathbb{Z}[x_{11}, . . . , x_{33}]$ into irreducible polynomials. L'anneau de polynômes $\mathbb{Z}[x_{11}, . . . , x_{33}]$ a une base appelée base duale canonique, et dont une quantification facilite l'étude des représentations du groupe quantique $U_q(\mathfrak{sl}3(\mathbb{C}))$. D'autre part, $\mathbb{Z}[x_{11}, . . . , x_{33}]$ admet une base issue de la base des monômes d'amas de l'algèbre amassée géométrique de type $D_4$. Nous montrons que ces deux bases sont égales. Ceci prolonge les travaux de Skandera et démontre une conjecture de Fomin et Zelevinsky. Ceci fournit également une factorisation explicite en polynômes irréductibles des éléments de la base duale canonique de $\mathbb{Z}[x_{11}, . . . , x_{33}]$ .

scholarly journals The cluster and dual canonical bases of Z[x(11), ..., x(33)] are equal

2010 ◽  
Vol Vol. 12 no. 5 (Combinatorics) ◽  
Author(s):  
Brendon Rhoades
Keyword(s):  
Quantum Group ◽  
Polynomial Ring ◽  
Geometric Model ◽  
Cluster Algebra ◽  
Canonical Basis ◽  
The Other ◽  
Monomial Basis ◽  
Type D ◽  
Dual Canonical Basis ◽  
International Audience

Combinatorics International audience The polynomial ring Z[x(11), ..., x(33)] has a basis called the dual canonical basis whose quantization facilitates the study of representations of the quantum group U-q(sl(3) (C)). On the other hand, Z[x(1 1), ... , x(33)] inherits a basis from the cluster monomial basis of a geometric model of the type D-4 cluster algebra. We prove that these two bases are equal. This extends work of Skandera and proves a conjecture of Fomin and Zelevinsky.

scholarly journals Arrangements of equal minors in the positive Grassmannian

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Miriam Farber ◽  
Alexander Postnikov
Keyword(s):  
Cluster Algebra ◽  
The Other ◽  
Eulerian Number ◽  
Other Hand ◽  
Totally Positive ◽  
Positive Matrices ◽  
International Audience ◽  
Separated Sets ◽  
Weakly Separated Sets ◽  
Totally Positive Matrices

International audience We discuss arrangements of equal minors in totally positive matrices. More precisely, we would like to investigate the structure of possible equalities and inequalities between the minors. We show that arrangements of equals minors of largest value are in bijection with <i>sorted sets</i>, which earlier appeared in the context of <i>alcoved polytopes</i> and Gröbner bases. Maximal arrangements of this form correspond to simplices of the alcoved triangulation of the hypersimplex; and the number of such arrangements equals the <i>Eulerian number</i>. On the other hand, we conjecture and prove in many cases that arrangements of equal minors of smallest value are exactly the <i>weakly separated sets</i>. Weakly separated sets, originally introduced by Leclerc and Zelevinsky, are closely related to the \textitpositive Grassmannian and the associated <i>cluster algebra</i>.

scholarly journals Atomic Bases and $T$-path Formula for Cluster Algebras of Type $D$

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Emily Gunawan ◽  
Gregg Musiker
Keyword(s):  
Cluster Algebra ◽  
Cluster Algebras ◽  
Combinatorial Proof ◽  
Expansion Formula ◽  
Type D ◽  
International Audience ◽  
T Path ◽  
Nous Utilisons

International audience We extend a $T$-path expansion formula for arcs on an unpunctured surface to the case of arcs on a once-punctured polygon and use this formula to give a combinatorial proof that cluster monomials form the atomic basis of a cluster algebra of type $D$. Nous généralisons une formule de développement en $T$-chemins pour les arcs sur une surface non-perforée aux arcs sur un polygone à une perforation. Nous utilisons cette formule pour donner une preuve combinatoire du fait que les monômes amassées constituent la base atomique d’une algèbre amassée de type $D$.

scholarly journals Coherent fans in the space of flows in framed graphs

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Vladimir I. Danilov ◽  
Alexander V. Karzanov ◽  
Gleb A. Koshevoy
Keyword(s):  
Simplicial Complex ◽  
Binary Relation ◽  
Directed Graph ◽  
The Other ◽  
Cluster Algebras ◽  
Linear Orders ◽  
International Audience ◽  
Acyclic Directed Graph ◽  
Extreme Rays ◽  
Space Of Flows

International audience Let $G=(V,E)$ be a finite acyclic directed graph. Being motivated by a study of certain aspects of cluster algebras, we are interested in a class of triangulations of the cone of non-negative flows in $G, \mathcal F_+(G)$. To construct a triangulation, we fix a raming at each inner vertex $v$ of $G$, which consists of two linear orders: one on the set of incoming edges, and the other on the set of outgoing edges of $v$. A digraph $G$ endowed with a framing at each inner vertex is called $framed$. Given a framing on $G$, we define a reflexive and symmetric binary relation on the set of extreme rays of $\mathcal F_+ (G)$. We prove that that the complex of cliques formed by this binary relation is a pure simplicial complex, and that the cones spanned by cliques constitute a unimodular simplicial regular fan $Σ (G)$ covering the entire $\mathcal F_+(G)$. Soit $G=(V,E)$ un graphe orientè, fini et acyclique. Nous nous intéressons, en lien avec l’étude de certains aspects des algèbres amassées, à une classe de triangulations du cône des flots positifs de $G, \mathcal F_+(G)$. Pour construire une triangulation, nous ajoutons une structure en chaque sommet interne $v$ de $G$, constituée de deux ordres totaux : l'un sur l'ensemble des arcs entrants, l'autre sur l'ensemble des arcs sortants de $v$. On dit alors que $G$ est structurè. On définit ensuite une relation binaire réflexive et symétrique sur l'ensemble des rayons extrêmes de $\mathcal F_+ (G)$. Nous démontrons que le complexe des cliques formè par cette relation binaire est un complexe simplicial pur, et que le cône engendré par les cliques forme un éventail régulier simplicial unimodulaire $Σ (G)$ qui couvre complètement $\mathcal F_+(G)$.

The centre of a quantum affine algebra at a root of unity

1994 ◽  
Vol 44 (11-12) ◽  
pp. 1091-1100
Author(s):  
Jens -Ulrik Holger Petersen
Keyword(s):  
Quantum Affine Algebra ◽  
Affine Algebra ◽  
Root Of Unity
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