Quantum Affine Lie Algebras, Casimir Invariants and Diagonalization of the Braid Generator

TL;DR

This paper constructs Casimir invariants for quantum affine Lie algebras, proves properties of the universal R-matrix, and diagonalizes the braid generator on tensor modules, providing explicit spectral decompositions.

Contribution

It introduces new Casimir invariants for quantum affine Lie algebras and establishes the diagonalization and spectral decomposition of the braid generator.

Findings

Eigenvalues of Casimir invariants are explicitly computed.

The universal R-matrix satisfies a conjugation relation involving the twist map.

The braid generator is shown to be diagonalizable with a spectral decomposition formula.

Abstract

Let $U_q(\hat{\cal G})$ be an infinite-dimensional quantum affine Lie algebra. A family of central elements or Casimir invariants are constructed and their eigenvalues computed in any integrable irreducible highest weight representation. These eigenvalue formulae are shown to absolutely convergent when the deformation parameter $q$ is such that $|q|>1$. 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, the braid generator is shown to be diagonalizable on arbitrary tensor product modules of integrable irreducible highest weight $U_q(\hat{\cal G})$-modules and a spectral decomposition formula for the braid generator is obtained which is the generalization of Reshetikhin's and Gould's forms to the present affine case. Casimir invariants acting on a specified module are also constructed and their eigenvalues, again absolutely convergent for $|q|>1$, computed by means of the spectral decomposition formula.