Factorial type I KMS states of Lie groups

TL;DR

This paper characterizes Gibbs states of finite-dimensional Lie groups as factorial type I KMS states, linking their properties to elliptic elements in the Lie algebra and providing a complete classification under certain conditions.

Contribution

It provides a complete characterization of Gibbs states as factorial type I KMS states for finite-dimensional Lie groups, connecting trace conditions to elliptic elements in the Lie algebra.

Findings

Gibbs states are precisely the factorial type I KMS states for these groups.

Trace conditions imply the generator is an elliptic element.

Complete classification of Lie algebras and representations satisfying the trace condition.

Abstract

Motivated by the study of KMS conditions for C*- or W*-dynamical systems defined by covariant unitary representations of topological groups, we consider Gibbs states of a finite-dimensional Lie group $G$ and prove that these are precisely the factorial type I KMS states. For an element $X\in \textbf{L}(G)$ and an irreducible unitary representation $\rho$ of $G$ satisfying $\text{tr}(e^{i\partial\rho(X)})=1$, the corresponding Gibbs state is defined as $\varphi(g)=\text{tr}(\rho(g)e^{i\partial\rho(X)})$. We prove that under the mild assumption that $\rho$ has discrete kernel, the condition $\text{tr}(e^{i\partial\rho(X)})<~\infty$ implies that the generator $X$ is an inner point of the set $\text{comp}(\mathfrak{g})$ of elliptic elements in $\mathfrak{g}$. This allows us to obtain a complete characterization of Lie algebras $\mathfrak{g}$, representations $\rho$ with discrete kernel and generators $X$ such that $\text{tr}(e^{i\partial\rho(X)})<\infty$.