Generalization of Weyl realization to a class of Lie superalgebras

This paper generalizes Weyl realization to a class of Lie superalgebras $\mathfrak{g}={\mathfrak{g}}_0\oplus {\mathfrak{g}}_1$ satisfying $[{\mathfrak{g}}_1,{\mathfrak{g}}_1]=\{0\}$. First, we give a novel proof of the Weyl realization of a Lie algebra ${\mathfrak{g}}_0$ by deriving a functional equation for the function that defines the realization. We show that this equation has a unique solution given by the generating function for the Bernoulli numbers. This method is then generalized to Lie superalgebras of the above type.