Quantum Sugawara operators in type A

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-08-28 DOI:10.1016/j.aim.2024.109907

Naihuan Jing , Ming Liu , Alexander Molev

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引用次数: 0

Abstract

The quantum Sugawara operators associated with a simple Lie algebra $g$ are elements of the center of a completion of the quantum affine algebra $U_{q} (\hat{g})$ at the critical level. By the foundational work of Reshetikhin and Semenov-Tian-Shansky (1990), such operators occur as coefficients of a formal Laurent series $ℓ_{V} (z)$ associated with every finite-dimensional representation V of the quantum affine algebra. As demonstrated by Ding and Etingof (1994), the quantum Sugawara operators generate all singular vectors in generic Verma modules over $U_{q} (\hat{g})$ at the critical level and give rise to a commuting family of transfer matrices. Furthermore, as observed by E. Frenkel and Reshetikhin (1999), the operators are closely related with the q-characters and q-deformed $W$ -algebras via the Harish-Chandra homomorphism.

We produce explicit quantum Sugawara operators for the quantum affine algebra of type A which are associated with primitive idempotents of the Hecke algebra and parameterized by Young diagrams. This opens a way to understand all the related objects via their explicit constructions. We consider one application by calculating the Harish-Chandra images of the quantum Sugawara operators. The operators act by scalar multiplication in the q-deformed Wakimoto modules and we calculate the eigenvalues by identifying them with the Harish-Chandra images.

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A 型量子菅原算子

与简单李代数 g 相关的量子菅原算子是量子仿射代数 Uq(gˆ)在临界水平上的完成中心的元素。根据 Reshetikhin 和 Semenov-Tian-Shansky (1990) 的奠基性工作，这类算子是与量子仿射代数的每个有限维表示 V 相关联的形式劳伦数列 ℓV(z) 的系数。正如 Ding 和 Etingof（1994）所证明的那样，量子菅原算子在临界水平上生成了 Uq（gˆ）上的泛 Verma 模块中的所有奇异向量，并产生了一个共转矩阵族。此外，正如 E. Frenkel 和 Reshetikhin（1999 年）所观察到的那样，这些算子通过 Harish-Chandra 同态与 q 字符和 q 变形 W 矩阵密切相关。我们为 A 型量子仿射代数提出了明确的量子菅原算子，这些算子与赫克代数的基元幂级数相关，并由杨图参数化。这开辟了一条通过其显式构造理解所有相关对象的途径。我们通过计算量子菅原算子的哈里什-钱德拉图像来考虑其中一个应用。这些算子在 q 变形的瓦基莫托模块中通过标量乘法起作用，我们通过将它们与哈里什-钱德拉图像相识别来计算特征值。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.