  For a quantum (many-body) system with dynamical symmetry described by a given
Lie group, we study the trace of exponential operators with complex
coefficients in one of the irreducible subspaces in terms of the boson
realization. By using this approach, for compact groups, we obtain the result
of the trace that is consistent with the well-known Weyl character formula. For
non-compact groups (with SU(1,1) as an application), convergent condition of
the trace is also obtained. This approach may be a powerful tool to study the
thermodynamics of quantum (many-body) systems in equilibrium state or
nonequilibrium processes.
