Generically-constrained quantum isotropy

TL;DR

This paper investigates the conditions under which the isotropy subgroup of a subspace in a quantum group representation acts trivially or abelianizes, showing that such subspaces form an open set in a certain algebraic topology.

Contribution

It generalizes classical rigidity results to the quantum setting, analyzing the openness of subspaces with trivial or abelianized isotropy actions in quantum group representations.

Findings

The set of subspaces with trivial isotropy action is Zariski open.

The results extend classical rigidity to quantum groups and quantum graphs.

Applicable to non-commutative versions of automorphism group triviality.

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).