  In this paper, we consider a class of linear quadratic extended mean field
games (MFGs) with common noises where the state coefficients and the cost
functional vary with the mean field term in a nonlinear way. Based on
stochastic maximum principle, solving the mean field game is transformed into
solving a conditional mean field forward-backward stochastic differential
equation (FBSDE). We first establish solvability for a type of (more general)
conditional mean field FBSDEs under monotonicity conditions. We further provide
some regularity results which lead to classical solutions for the associated
master equations. In particular, the linear quadratic extended mean field game
is solved and classical solution for (extended mean field game) master equation
is obtained.
