The real sphere $S^{ N-1}_\mathbb R$ appears as increasing union, over $d\in\{ 1,...,N\}$, of its "polygonal" versions $S^{ N-1,d-1}_\mathbb R=\{ x\in S^{ N-1}_\mathbb R|x_{ i_0}... x_{ i_d}=0,\forall i_0,...,i_d\ { \rm distinct}\}$. Motivated by general classification questions for the undeformed noncommutative spheres, smooth or not, we study here the quantum isometries of $S^{ N-1,d-1}_\mathbb R$, and of its various noncommutative analogues, obtained via liberation and twisting. We discuss as well a complex version of these results, with $S^{ N-1}_\mathbb R$ replaced by the complex sphere $S^{ N-1}_\mathbb C$.
Categories: math.OA math.QA