Notes on Chevalley Groups and Root Category II: Compact Lie Groups and Representations

TL;DR

This paper explores the structure of compact Lie groups and their representations using root categories, connecting classical theories with quantum group frameworks to recover fundamental theorems.

Contribution

It introduces a method to derive classical compact Lie group theories from root categories and quantum group structures, extending previous work.

Findings

Defined compact real forms of complex semisimple Lie algebras

Established maximal compact subgroups of Chevalley groups over d6

Reconstructed classical theorems like Peter-Weyl and Plancherel

Abstract

This paper is a continuation of [5]. Using the root categories, we define the compact real forms of the complex semisimple Lie algebras, and maximal compact subgroups of the Chevalley groups over $\mathbb{C}$. In [7], Lusztig used the modified quantum group $\dot{\mathbf{U}}$ and its canonical basis to obtain the reductive group and its coordinate ring $\mathbf{O}_A$, in particular the tensor product decomposition of $\mathbf{O}_A$. By combining these two kinds of structures, we explore in this paper how the classical theory of the compact Lie groups, such as Peter-Weyl theorem and Plancherel theorem, can be recovered completely.