  We carefully analyze the supersymmetry algebra of closed strings and open
strings in a type IIB plane wave background. We use eight component chiral
spinors, SO(8) Majorana-Weyl spinors, in light-cone gauge to provide a useful
basis of string field theory calculation in the plane wave. We consider the two
classes of D-branes, $D_\pm$-branes, and give a worldsheet derivation of
conserved supercurrents for all half BPS D-branes preserving 16 supersymmetries
in the type IIB plane wave background. We exhaustively provide the
supersymmetry algebra of the half BPS branes as well. We also point out that
the supersymmetry algebra distinguishes the two SO(4) directions with relative
sign which is consistent with the Z_2 symmetry of the string action.
