Generically constrained quantum isotropy

Abstract

Let $V$ be a finite-dimensional unitary representation of a compact quantum group $\mathrm{G}$ and denote by ${\mathrm{G}}_W$ the isotropy subgroup of a linear subspace $W\le V$ regarded as a point in the Grassmannian $\mathbb{G}(V)$. We show that the space of those $W\in \mathbb{G}(V)$ for which ${\mathrm{G}}_W$ acts trivially on $W$ (or $V$) is open in the Zariski topology of the Weil restriction ${\mathrm{Res}}_{\mathbb{C}/\mathbb{R}}\mathbb{G}(V)$. More generally, this holds for the space of $W$ for which (a) the ${\mathrm{G}}_W$-action factors through its abelianization or (b) the summands of the ${\mathrm{G}}_W$-representation on $W$ (or $V$) are otherwise dimensionally constrained. The results generalize analogous classical generic rigidity statements useful in establishing the triviality of the classical automorphism groups of random quantum graphs in the matrix algebra $M_n$ and can be put to similar use in fully non-commutative versions of those results (quantum graphs, quantum groups).