Hom-Lie groups of a class of Hom-Lie algebra

In this paper, the definition of Hom-Lie groups is given and one conntected component of Lie group $GL(V)$, which is not a subgroup of $GL(V)$, is a Hom-Lie group. More, we proved that there is a one-to-one relationship between Hom-Lie groups and Hom-Lie algebras $(\text{gl}(V),[\cdot ,\cdot {]}_{\beta },{\mathrm{A}\mathrm{d}}_{\beta })$. Next, we also proved that if there is a Hom-Lie group homomorphism, then, there is a morphism between their Hom-Lie algebras. Last, as an application, we use these results on Toda lattice equation.