Exponential map and invariant form on generalized Lie groups

In this paper, by definition of exponential map of the Lie groups the concept of exponential map of generalized Lie groups is introduced. This has a powerful generalization to generalized Lie groups which takes each line through the origin to an order product of some one-parameter subgroup. We show that the exponential map is a ????- map. Also, we prove some important properties of the exponential map for generalized Lie groups. Under the identification, it is shown that the derivative of the exponential map is the identity map. One of the most powerful applications of these exponential maps is to define generalized adjoint representation of a top space, so we show that this representation is a ????- map. Finally, invariant forms are introduced on a generalized Lie group. We prove that every left invariant ??-form are introduced on a generalized Lie group ?? with the finite number of identity elements is ????. At the end of this paper, for compact connected generalized Lie group ?? with the finite number of identity elements and dimension ??, we show that every left invariant ??-form on ?? is right invariant ??-form.