Lie Superalgebras and Super-Representations: Examples and Counterexamples

The coordinate forms ${\varphi }_1,{\varphi }_2$ are linear by [F5]. For any $\varphi \in S_{-}$ and $(z_1,z_2)\in S_{+}$, linearity gives $\varphi (z_1,z_2)=z_1\varphi (u_1)+z_2\varphi (u_2)$, hence $\varphi =\varphi (u_1){\varphi }_1+\varphi (u_2){\varphi }_2$; evaluating at $u_1,u_2$ proves their independence. Thus $u_1,u_2$ and ${\varphi }_1,{\varphi }_2$ are bases of the two free $\mathbb{C}$-modules. By [F6], the scalar ring is commutative; [F4] then makes their four elementary tensors a basis of $S_{+}\otimes S_{-}$ and hence of the translation space.

The bracket formula in [F1] gives $[Q_{\alpha },{\bar{Q}}_{\dot{\beta }}]=P_{\alpha \dot{\beta }}$. Because both entries are odd, graded skew symmetry makes this bracket symmetric; in a graded associative representation [F3] identifies it with the anticommutator. Same-chirality odd brackets vanish and translations commute by the definition in [F1].

Let ${\sigma }^0=I$, ${\sigma }^1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, ${\sigma }^2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)$, and ${\sigma }^3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$. Direct multiplication gives $\mathrm{tr}({\sigma }^{\mu }{\sigma }^{\nu })=2{\delta }_{\mu \nu }$, so ${\tau }^{\mu }={\sigma }^{\mu }\mathrm{/}2$ is trace-dual to the four ${\sigma }^{\nu }$. The trace-pairing identity makes the ${\sigma }^{\mu }$ linearly independent; since a $2\times 2$ matrix has four entries, they form a basis of $M_2(\mathbb{C})$. Regard $P=(P_{\alpha \dot{\beta }})$ as a matrix with entries in $V$, and define $P_{\mu }=(2i{)}^{-1}\mathrm{tr}({\tau }^{\mu }P)$. The dual-basis expansion gives $P={\sum }_{\mu }{\sigma }^{\mu }\mathrm{tr}({\tau }^{\mu }P)=2i{\sum }_{\mu }{\sigma }^{\mu }{P}_{\mu }$. This is an invertible coordinate change followed by multiplication by the nonzero scalar $(2i{)}^{-1}$, so the $P_{\mu }$ form a basis of translations and the displayed normalization follows. The dotted bar on $\bar{Q}$ is a chiral label, not complex conjugation. This algebraic normalization supplies no real form, positive-energy representation, or field-theoretic off-shell closure.

By [F3] and [F1], the defining relation is ${\iota }_{\mathfrak{a}}(e{)}^2=0$; the tensor quotient is generated by $1$ and ${\iota }_{\mathfrak{a}}(e)$, so every word reduces to a linear combination of them.

By [F4], $1,e$ are linearly independent in the quotient and span it, so $U(\mathfrak{a})=\mathbb{C}\oplus \mathbb{C}e$ as a vector space with the stated multiplication.

The subspace $\mathbb{C}w$ is a graded submodule: it is preserved by the diagonal operators $h_1,h_2$, while $ew=fw=0$. It is simple because it is one-dimensional and nonzero. The quotient line generated by $v+\mathbb{C}w$ is also simple: $h_1,h_2,e,f$ all act by zero on it.

The only graded complement to $\mathbb{C}w$ would have even part $\mathbb{C}v$ and zero odd part, since $K(0,0{)}_{\bar{0}}=\mathbb{C}v$ and $K(0,0{)}_{\bar{1}}=\mathbb{C}w$. But $fv=w\mathrm{\notin }\mathbb{C}v$, so this complement is not a submodule. Thus the exact sequence with submodule $\mathbb{C}w$ and quotient $K(0,0)\mathrm{/}\mathbb{C}w$ does not split.

Any nonzero graded submodule containing an even vector contains $v$, hence also $fv=w$, and equals the whole module; a nonzero graded submodule with zero even part is exactly $\mathbb{C}w$. Thus $\mathbb{C}w$ is the only simple graded submodule, and its direct sums cannot equal the $1\mathrm{\mid }1$-dimensional $K(0,0)$. By [F4], this module is not semisimple; [F5] confirms it is an instance covered by the refuted claim.

By [F1], ${\mathfrak{n}}^{+}=\mathbb{C}e$ and $\mathfrak{h}=\mathbb{C}{h}_1\oplus \mathbb{C}{h}_2$. The vector $v$ is nonzero, even, has the displayed $\mathfrak{h}$-weight, and is killed by $e$, so [F2] and [F4] identify $K(a,b)$ as the stated highest-weight supermodule. Its only nonzero proper graded subspaces are $\mathbb{C}v$ and $\mathbb{C}w$, since its even and odd parts are one-dimensional. Neither is invariant when $a+b\mathrm{\ne }0$: $fv=w\mathrm{\notin }\mathbb{C}v$, while $ew=(a+b)v\mathrm{\notin }\mathbb{C}w$. By [F6], it is simple for a typical parameter.

If $a+b=0$, then $ew=0$, $fw=0$, and $h_1,h_2$ preserve $\mathbb{C}w$, so $\mathbb{C}w$ is a nonzero proper graded submodule. By [F6], $K(a,b)$ is not simple; by [F5], the parameter is atypical.