ترغب بنشر مسار تعليمي؟ اضغط هنا

Simply-laced root systems arising from quantum affine algebras

208   0   0.0 ( 0 )
 نشر من قبل Euiyong Park
 تاريخ النشر 2020
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

Let $U_q(mathfrak{g})$ be a quantum affine algebra with an indeterminate $q$ and let $mathscr{C}_{mathfrak{g}}$ be the category of finite-dimensional integrable $U_q(mathfrak{g})$-modules. We write $mathscr{C}_{mathfrak{g}}^0$ for the monoidal subcategory of $mathscr{C}_{mathfrak{g}}$ introduced by Hernandez-Leclerc. In this paper, we associate a simply-laced finite type root system to each quantum affine algebra $U_q(mathfrak{g})$ in a natural way, and show that the block decompositions of $mathscr{C}_{mathfrak{g}}$ and $mathscr{C}_{mathfrak{g}}^0$ are parameterized by the lattices associated with the root system. We first define a certain abelian group $mathcal{W}$ (resp. $mathcal{W}_0$) arising from simple modules of $ mathscr{C}_{mathfrak{g}}$ (resp. $mathscr{C}_{mathfrak{g}}^0$) by using the invariant $Lambda^infty$ introduced in the previous work by the authors. The groups $mathcal{W}$ and $mathcal{W}_0$ have the subsets $Delta$ and $Delta_0$ determined by the fundamental representations in $ mathscr{C}_{mathfrak{g}}$ and $mathscr{C}_{mathfrak{g}}^0$ respectively. We prove that the pair $( mathbb{R} otimes_mathbb{Z} mathcal{W}_0, Delta_0)$ is an irreducible simply-laced root system of finite type and the pair $( mathbb{R} otimes_mathbb{Z} mathcal{W}, Delta) $ is isomorphic to the direct sum of infinite copies of $( mathbb{R} otimes_mathbb{Z} mathcal{W}_0, Delta_0)$ as a root system.



قيم البحث

اقرأ أيضاً

We introduce and investigate new invariants on the pair of modules $M$ and $N$ over quantum affine algebras $U_q(mathfrak{g})$ by analyzing their associated R-matrices. From new invariants, we provide a criterion for a monoidal category of finite-dim ensional integrable $U_q(mathfrak{g})$-modules to become a monoidal categorification of a cluster algebra.
Let $U_q(mathfrak{g})$ be a quantum affine algebra of arbitrary type and let $mathcal{C}_{mathfrak{g}}$ be Hernandez-Leclercs category. We can associate the quantum affine Schur-Weyl duality functor $F_D$ to a duality datum $D$ in $mathcal{C}_{mathfr ak{g}}$. We introduce the notion of a strong (complete) duality datum $D$ and prove that, when $D$ is strong, the induced duality functor $F_D$ sends simple modules to simple modules and preserves the invariants $Lambda$ and $Lambda^infty$ introduced by the authors. We next define the reflections $mathcal{S}_k$ and $mathcal{S}^{-1}_k$ acting on strong duality data $D$. We prove that if $D$ is a strong (resp. complete) duality datum, then $mathcal{S}_k(D)$ and $mathcal{S}_k^{-1}(D)$ are also strong (resp. complete ) duality data. We finally introduce the notion of affine cuspidal modules in $mathcal{C}_{mathfrak{g}}$ by using the duality functor $F_D$, and develop the cuspidal module theory for quantum affine algebras similarly to the quiver Hecke algebra case.
146 - Jie Xiao , Han Xu , Minghui Zhao 2021
Let $textbf{U}^+$ be the positive part of the quantum group $textbf{U}$ associated with a generalized Cartan matrix. In the case of finite type, Lusztig constructed the canonical basis $textbf{B}$ of $textbf{U}^+$ via two approaches. The first one is an elementary algebraic construction via Ringel-Hall algebra realization of $textbf{U}^+$ and the second one is a geometric construction. The geometric construction of canonical basis can be generalized to the cases of all types. The generalization of the elementary algebraic construction to affine type is an important problem. We give several main results of algebraic constructions to the affine canonical basis in this ariticle. These results are given by Beck-Nakajima, Lin-Xiao-Zhang, Xiao-Xu-Zhao, respectively.
106 - Shoumin Liu 2019
The Morita equivalences of classical Brauer algebras and classical Birman-Murakami-Wenzl algebras have been well studied. Here we study the Morita equivalence problems on these two kinds of algebras of simply-laced type, especially for them with the generic parameters. We show that Brauer algebras and Birman-Murakami-Wenzl algebras of simply-laced type are Morita equivalent to the direct sums of some group algebras of Coxeter groups and some Hecke algebras of some Coxeter groups, respectively.
In this paper, extending the idea presented by M. Takeuchi in [13], we introduce the notion of partial matched pair $(H,L)$ involving the concepts of partial action and partial coaction between two Hopf algebras $H$ and $L$. Furthermore, we present n ecessary conditions for the corresponding bismash product $L# H$ to generate a new Hopf algebra and, as illustration, a family of examples is provided.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا