量子仿射代数Uq(sln) 的范畴化

数学物理学报 ›› 2013, Vol. 33 ›› Issue (3): 431-465.

量子仿射代数U_q(sl_n) 的范畴化

任磊|常文静|王娜|王志玺|吴可|杨紫峰

首都师范大学数学科学学院北京 |100048

收稿日期:2011-10-23 修回日期:2013-02-01 出版日期:2013-06-25 发布日期:2013-06-25
基金资助:
国家自然科学基金(10975102, 11031005, 10871135, 10871227)和北京市创新团队项目(201007107) 资助

The Categorification of Quantum Affine Algebra U_q(sl_n)

REN Lei, CHANG Wen-Jing, WANG Na, WANG Zhi-Xi, WU Ke, YANG Zi-Feng

School of Mathematical Science, Capital Normal University, Beijing 100048

Received:2011-10-23 Revised:2013-02-01 Online:2013-06-25 Published:2013-06-25
Supported by:
国家自然科学基金(10975102, 11031005, 10871135, 10871227)和北京市创新团队项目(201007107) 资助

摘要/Abstract

摘要：

用弦图构造了与Lusztig的量子群U=U_q(sl_n) 对应的~2 -范畴U和U. 其中U是U的Karoubi包络. 通过构造U的2 -表示, 可以证明 U的Grothendieck环K₀(U)同构于代数_AU, 其中U是U 的幂等变形, 而_AU是U的整形式. 由此得到的范畴U就是量子仿射代数U_q(sl_n)的范畴化.

关键词: 量子仿射代数, 范畴化, 2 -表示

Abstract:

We construct 2-categories U and U using string diagrams corresponding to Lusztig's quantum group U=U_q(sln). Here U is the Karoubi envelope of U. By constructing the 2-representation of U, we also prove that the Grothendieck ring K₀(U) of U is isomorphic to the algebra _AU, where U is the idempotented modification of U and _AU is the integral form of U. We obtain that the category U could be considered as the categorification of quantum affine algebra U_q(sl_n).

Key words: Quantum affine algebra, Categorification, 2-representation

中图分类号:

81R10

任磊, 常文静, 王娜, 王志玺, 吴可, 杨紫峰. 量子仿射代数U_q(sl_n) 的范畴化[J]. 数学物理学报, 2013, 33(3): 431-465.

REN Lei, CHANG Wen-Jing, WANG Na, WANG Zhi-Xi, WU Ke, YANG Zi-Feng. The Categorification of Quantum Affine Algebra U_q(sl_n)[J]. Acta mathematica scientia,Series A, 2013, 33(3): 431-465.

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