  In this paper, a class of time inconsistent linear quadratic optimal control
problems of mean-field stochastic differential equations (SDEs) is considered
under Markovian framework. Open-loop equilibrium controls and their particular
closed-loop representations are introduced and characterized via variational
ideas. Several interesting features are revealed and a system of coupled
Riccati equations is derived. In contrast with the analogue optimal control
problems of SDEs, the mean-field terms in state equation, which is another
reason of time inconsistency, prompts us to define above two notions in new
manners. An interesting result, which is almost trivial in the counterpart
problems of SDEs, is given and plays significant role in the previous
characterizations. As application, the uniqueness of open-loop equilibrium
controls is discussed.
