Dirac reduction algebra

There is a homomorphism of associative superalgebras from the enveloping algebra of the orthosymplectic Lie superalgebra $\mathfrak{o}\mathfrak{s}\mathfrak{p}(1\mathrm{\mid }2)$ to the Weyl-Clifford superalgebra $W(2n\mathrm{\mid }n)$ with $2n$ even Weyl algebra generators and $n$ odd Clifford algebra generators. Under this homomorphism, the positive odd root vector $x\in \mathfrak{o}\mathfrak{s}\mathfrak{p}(1\mathrm{\mid }2)$ is sent to the Dirac operator ${\gamma }^{\mu }{\mathrm{\partial }}_{\mu }\in W(2n\mathrm{\mid }n)$ and generates a left ideal $I$. The corresponding reduction (super)algebra, denoted $Z_n$, is the normalizer of $I$ in $W(2n\mathrm{\mid }n)$ modulo $I$. By construction, $Z_n$ acts on the space of all Clifford algebra-valued polynomial solutions to the (massless) Dirac equation. In this paper, we find a complete presentation of (a localization of) this so-termed Dirac reduction algebra. Furthermore, we use the Dirac reduction algebra to generate all polynomial solutions to the Dirac equation in $n$-dimensional flat spacetime.