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 types A and C, and the direct method for types A, D, and B (level 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 typesAandC, and the direct method for typesA, D, andB(level 2).

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 Bar-invariant bases of the quantum cluster algebra of type A 2 (2)

2011 ◽  
Vol 61 (4) ◽  
pp. 1077-1090 ◽  
Author(s):  
Xueqing Chen ◽  
Ming Ding ◽  
Jie Sheng
Keyword(s):  
Cluster Algebra ◽  
Type A ◽  
Quantum Cluster Algebra ◽  
Quantum Cluster

scholarly journals TWO DESCRIPTIONS OF THE QUANTUM AFFINE ALGEBRA Uv() VIA HALL ALGEBRA APPROACH

2011 ◽  
Vol 54 (2) ◽  
pp. 283-307 ◽  
Author(s):  
IGOR BURBAN ◽  
OLIVIER SCHIFFMANN
Keyword(s):  
Integral Form ◽  
Derived Categories ◽  
Quantum Affine Algebra ◽  
Hall Algebra ◽  
Affine Algebra ◽  
Coherent Sheaves ◽  
Enveloping Algebra ◽  
Algebra Approach ◽  
Quantized Enveloping Algebra

AbstractWe compare the reduced Drinfeld doubles of the composition subalgebras of the category of representations of the Kronecker quiver and the category of coherent sheaves on ℙ1. Using this approach, we show that the Drinfeld–Beck isomorphism for the quantized enveloping algebra Uv() is a corollary of an equivalence between the derived categories Db(Rep()) and Db(Coh(ℙ1)). This technique allows to reprove several results on the integral form of Uv().

Sign in / Sign up

Export Citation Format

Share Document