  Motivated by recent development in quantum fidelity and fidelity
susceptibility, we study relations among Lie algebra, fidelity susceptibility
and quantum phase transition for a two-state system and the
Lipkin-Meshkov-Glick model. We get the fidelity susceptibility for SU(2) and
SU(1,1) algebraic structure models. From this relation, the validity of the
fidelity susceptibility to signal for the quantum phase transition is also
verified in these two systems. At the same time, we obtain the geometric phase
in these two systems in the process of calculating the fidelity susceptibility.
In addition, the new method of calculating fidelity susceptibility has been
applied to explore the two-dimensional XXZ model and the Bose-Einstein
condensate(BEC).
