Finite-dimensional Lie supermodule categories need not be semisimple

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.