= Cubic-root quantum-plane socle
{title2=$\operatorname{soc}(k_q^3[x,y])=\langle x^3,y^3\rangle$}
At a primitive cube <root of unity>, the cubic homogeneous <module> has <basis> $x^3,x^2y,xy^2,y^3$. Its only <simple modules> occurring as <submodules> are the one-dimensional subspaces of $\langle x^3,y^3\rangle$, on which $K=1$ and $E=F=0$. Its quotient by this two-dimensional <submodule> is simple, but it has no simple complement: any lift of the highest $K$ <weight space> contains $x^2y$, whose $E$ image is $x^3\ne0$. Thus the cubic <module> is not a <semisimple module>.
Back to article page
