Generalization of Weyl realization to a class of Lie superalgebras

TL;DR

This paper extends the Weyl realization method from Lie algebras to a specific class of Lie superalgebras, providing a new proof and a functional equation approach involving Bernoulli numbers.

Contribution

It introduces a novel proof technique for Weyl realization and generalizes it to certain Lie superalgebras with a unique solution linked to Bernoulli numbers.

Findings

Derived a functional equation for the realization function.

Proved the uniqueness of the solution involving Bernoulli numbers.

Generalized the Weyl realization to Lie superalgebras of a specific type.

Abstract

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.