Quantized Affine Lie Algebras and Diagonalization of Braid Generators

TL;DR

This paper proves a key conjugation relation for the universal R-matrix of quantized affine Lie algebras, leading to diagonalization of braid generators and spectral decomposition formulas for these algebraic structures.

Contribution

It establishes the conjugation relation for the universal R-matrix and generalizes spectral decomposition formulas for braid generators in affine quantum groups.

Findings

Braid generators are diagonalizable on tensor product modules.

Spectral decomposition formulas are derived for affine cases.

Eigenvalues of Casimir invariants are computed using the new formulas.

Abstract

Let $U_q(\hat{\cal G})$ be a quantized affine Lie algebra. It is proven that the universal R-matrix $R$ of $U_q(\hat{\cal G})$ satisfies the celebrated conjugation relation $R^\dagger=TR$ with $T$ the usual twist map. As applications, braid generators are shown to be diagonalizable on arbitrary tensor product modules of integrable irreducible highest weight $U_q(\hat{\cal G})$-module and a spectral decomposition formula for the braid generators is obtained which is the generalization of Reshetikhin's and Gould's forms to the present affine case. Casimir invariants are constructed and their eigenvalues computed by means of the spectral decomposition formula. As a by-product, an interesting identity is found.