q-deformation of the square white noise Lie algebra

For $q\in (0,1)$, the $q$-deformation of the square white noise Lie algebra is introduced using the $q$-calculus. A representation of this Lie algebra is given, using the $q$-derivative (or Jackson derivative) and the multiplication operator. The free square white noise Lie algebra is defined. Moreover, its representation on the Hardy space is given.