Category O is not semisimple at the rank-one parameter c=1/2

By [F2] and [F5], $y{x}^n=(n-c(1-(-1{)}^n))x^{n-1}$ for every $n\ge 1$, while $y1=0$. At $c=1\mathrm{/}2$, this is $n{x}^{n-1}$ for even $n$ and $(n-1)x^{n-1}$ for odd $n$. In particular, $yx=0$.

The subspace $I=x\mathbb{C}[x]$ is nonzero and proper. Multiplication by $x$ preserves $I$, $s(x^n)=(-1{)}^n{x}^n$ preserves $I$, and step 1.1 shows $y(x^n)\in I$ for every $n\ge 1$: for $n=1$ it is zero, while for $n\ge 2$ its polynomial degree is $n-1\ge 1$. By [F3], this is the polynomial standard module; [F4] places it and its submodules in ${\mathcal{O}}_c$.

The quotient $\mathbb{C}[x]\mathrm{/}I$ is one dimensional; $x$ and $y$ act by zero and $s$ acts trivially, so it is simple. If the displayed exact sequence split, its complement would be a one-dimensional submodule $J$ mapping isomorphically to the quotient. A generator $f\in J$ would have nonzero constant term; rescale so $f(0)=1$. Since $xJ\subseteq J$, there would be a scalar $\lambda$ with $xf=\lambda f$, or $(x-\lambda )f=0$. The polynomial ring $\mathbb{C}[x]$ is an integral domain and both factors are nonzero, a contradiction. The sequence is nonsplit, so ${\mathcal{O}}_c$ is not semisimple.