In this paper we have introduced new invariant geometric objects in the homogeneous spaces of complex, dual and double numbers for the principal group $SL(2;\mathbb{R})$, in the Klein's Erlangen Program. We have considered the action as the Möbius action and have taken the spaces as the spaces of complex, dual and double numbers. Some new decompositions of $SL(2;\mathbb{R})$ have been used.