This paper investigates the solvability of the Sylvester matrix equation $AX-XB=C$ with respect to left semi-tensor product. Firstly, we discuss the matrix-vector equation $AX-XB=C$ under semi-tensor product. A necessary and sufficient condition for the solvability of the matrix-vector equation and specific solving methods are studied and given. Based on this, the solvability of the matrix equation $AX-XB=C$ under left semi-tensor product is discussed. Finally, several examples are presented to illustrate the efficiency of the results.