Differential Geometry of the Lie algebra of the quantum plane

We present a differential calculus on the extension of the quantum plane obtained considering that the (bosonic) generator $x$ is invertible and furthermore working polynomials in $\ln x$ instead of polynomials in $x$. We call quantum Lie algebra to this extension and we obtain its Hopf algebra structure and its dual Hopf algebra. Differential geometry of the quantum Lie algebra of the extended quantum plane and its Hopf algebra structure is obtained. Its dual Hopf algebra is also given.