2023/01/09 by Tobias Simon, Simon, Tobias
Materials Science · Physics and Astronomy · #FOS: Mathematics #Organic and Molecular Conductors Research #Quantum chaos and dynamical systems #Quantum many-body systems #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2301.03444
openalex publication_date 2023/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
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∈ L(G) and an irreducible unitary representation ρ of G satisfying tr(ei∂ρ(X))=1, the corresponding Gibbs state is defined as φ(g)=tr(ρ(g)ei∂ρ(X)). We prove that under the mild assumption that ρ has discrete kernel, the condition tr(ei∂ρ(X))<~∞ implies that the generator X is an inner point of the set comp(\mathfrakg) of elliptic elements in \mathfrakg. This allows us to obtain a complete characterization of Lie algebras \mathfrakg, representations ρ with discrete kernel and generators X such that tr(ei∂ρ(X))<∞.