Rigidity questions for real half-classical manifolds

Let $X$ be a noncommutative real compact algebraic manifold, in the sense that $C(X)=C^{\ast }(x_1,\dots ,x_N\mathrm{\mid }{x}_i=x_i^{\ast },f_{\alpha }(x_1,\dots ,x_N)=0)$, with $f_{\alpha }\in \mathbb{R}<x_1,\dots ,x_N>$. Associated to $X$ are its classical version $X^{\times }$, obtained via the relations $ab=ba$, and its half-classical version $X^{\ast }$, obtained via the relations $abc=cba$. We discuss here some general questions regarding the inclusions $X^{\times }\subset X^{\ast }\subset X$, and notably the comparison of the corresponding quantum isometry groups. Our main results concern the half-classical case, $X=X^{\ast }$, and more specifically, the case of the submanifolds $X\subset S_{\mathbb{R},\ast }^{N-1}$.