  In recent years, the study of slice monogenic functions has attracted more
and more attention in the literature. In this paper, an extension of the
well-known Dirac operator is defined which allows to establish the Lie
superalgebra structure behind the theory of slice monogenic functions.
Subsequently, an inner product is defined corresponding to this slice Dirac
operator and its polynomial null-solutions are determined. Finally, analogues
of the Hermite polynomials and Hermite functions are constructed in this
context and their properties are studied.
