Complexified four-dimensional N=1 super-Poincare bracket data

Let $S_{+}={\mathbb{C}}^2$, $S_{-}=({\mathbb{C}}^2{)}^{\ast }$, where the algebraic dual is defined in Linear functionals and the algebraic dual $V^{\ast }=\mathcal{L}(V,F)$, and set $V=S_{+}{\otimes }_{\mathbb{C}}{S}_{-}$. By Every field is a commutative ring with $1\mathrm{\ne }0$; it is an integral domain, and it is a commutative division ring, $\mathbb{C}$ is a commutative ring; use the scalar actions to regard $S_{+}$ as a right module and $S_{-}$ as a left module. Use their module tensor product from The tensor product $M{\otimes }_{R}N$ from the additive group underlying the free $\mathbb{Z}$-module on $M\times N$, elementary tensors, and finite tensor sums; its universal property is Universal property of the tensor product for balanced maps into abelian groups, and its unique $\mathbb{C}$-module structure is given by Over a commutative ring, $M{\otimes }_{R}N$ is an $R$-module with $r(m\otimes n)=(rm)\otimes n=m\otimes (rn)$. Thus $V$ is a complex vector space, every tensor is a finite sum of elementary tensors, and bilinear formulas on $S_{+}\times S_{-}$ descend to linear maps on $V$.

Let $\mathfrak{l}=\mathfrak{s}\mathfrak{l}(S_{+})\oplus \mathfrak{s}\mathfrak{l}({\mathbb{C}}^2)\mathrm{.}$ Using the standard basis of $S_{+}$, both summands are copies of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$, the traceless complex matrix Lie algebra with commutator from Classical complex matrix Lie algebras. Use the natural action of the first factor on $S_{+}$, the dual action $B\cdot \varphi =-\varphi B$ of the second factor on $S_{-}$, and the tensor action on $V=S_{+}\otimes S_{-}$: $(A,B)\cdot (u\otimes \varphi )=Au\otimes \varphi -u\otimes \varphi B\mathrm{.}$ Let the even part be ${\mathfrak{g}}_{\bar{0}}=\mathfrak{l}\oplus V$, with bracket $[(L,v),(L^{\mathrm{\prime }},v^{\mathrm{\prime }})]=([L,L^{\mathrm{\prime }}],L\cdot v^{\mathrm{\prime }}-L^{\mathrm{\prime }}\cdot v),$ and let the odd part be ${\mathfrak{g}}_{\bar{1}}=S_{+}\oplus S_{-}$. The even action on the odd part is $(L,v)\cdot s=L\cdot s$, so translations $V$ act trivially. Define the odd bracket by $[u,u^{\mathrm{\prime }}]=0,[\varphi ,{\varphi }^{\mathrm{\prime }}]=0,[u,\varphi ]=[\varphi ,u]=u\otimes \varphi$ for $u,u^{\mathrm{\prime }}\in S_{+}$ and $\varphi ,{\varphi }^{\mathrm{\prime }}\in S_{-}$. Extend all brackets bilinearly and by graded skew symmetry. This is the complexified spinor-form bracket data for the four-dimensional $N=1$ super-Poincare algebra. A separate theorem proves that the data satisfy the Lie-superalgebra axioms. The ordinary real form, its unitary representations, and a field-theoretic realization are not included in this definition.