Quantized Affine Lie Algebras and Diagonalization of Braid Generators

Abstract

Let $U_q(\hat{\mathcal{G}})$ be a quantized affine Lie algebra. It is proven that the universal R-matrix $R$ of $U_q(\hat{\mathcal{G}})$ satisfies the celebrated conjugation relation $R^{†}=TR$ 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 $U_q(\hat{\mathcal{G}})$-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.

11 pages (minor error corrected)

References in corpus (1)

• On Universal $R$-Matrix for Quantized Nontwisted Rank 3 Affine Lie Algebras

Cited by in corpus (2)

• On the Construction of Trigonometric Solutions of the Yang-Baxter Equation
• Quantum Affine Lie Algebras, Casimir Invariants and Diagonalization of the Braid Generator