Complex and biHermitian structures on four dimensional real Lie algebras

We give a new method for calculation of complex and biHermitian structures on low dimensional real Lie algebras. In this method, using non-coordinate basis, we first transform the Nijenhuis tensor field and biHermitian structure relations on Lie groups to the tensor relations on their Lie algebras. Then we use adjoint representation for writing these relations in the matrix form; in this manner by solving these matrix relations and using automorphism groups of four dimensional real Lie algebras we obtain and classify all complex and biHermitian structures on four dimensional real Lie algebras.