  We investigate the boundedness of Fourier multipliers on a compact Lie group
when acting on Triebel-Lizorkin spaces. Criteria are given in terms of the
H\"ormander-Mihlin-Marcinkiewicz condition. In our analysis, we use the
difference structure of the unitary dual of a compact Lie group. Our results
cover the sharp H\"ormander-Mihlin theorem on Lebesgue spaces and also other
historical results on the subject.
