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Generically-constrained quantum isotropy

Alexandru Chirvasitu

TL;DR

The paper proves that the spaces of subspaces $W$ where the isotropy subgroup $\mathrm{G}_W$ acts trivially on $W$ (or on $V$) or through its abelianization form a Zariski-open set in the Weil-restricted Grassmannian, providing a quantum analogue of classical generic-rigidity results. It develops a framework using incidence correspondences and Chevalley-type arguments within the Tannaka reconstruction/coend formalism to show openness and density of these sets in the quantum group setting, and extends the results to $\ell$-constrained representations and abelian quotients. A parallel result for operator algebras shows that generating $n$-tuples are Zariski-open and dense in $A^n$ whenever $A$ is separable, with a specialization to generating self-adjoint pairs in finite-dimensional C$^*$-algebras. The findings provide algebraic-geometric tools for establishing generic rigidity and triviality of automorphism structures in quantum graphs and non-commutative analogues.

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).

Generically-constrained quantum isotropy

TL;DR

The paper proves that the spaces of subspaces where the isotropy subgroup acts trivially on (or on ) or through its abelianization form a Zariski-open set in the Weil-restricted Grassmannian, providing a quantum analogue of classical generic-rigidity results. It develops a framework using incidence correspondences and Chevalley-type arguments within the Tannaka reconstruction/coend formalism to show openness and density of these sets in the quantum group setting, and extends the results to -constrained representations and abelian quotients. A parallel result for operator algebras shows that generating -tuples are Zariski-open and dense in whenever is separable, with a specialization to generating self-adjoint pairs in finite-dimensional C-algebras. The findings provide algebraic-geometric tools for establishing generic rigidity and triviality of automorphism structures in quantum graphs and non-commutative analogues.

Abstract

Let be a finite-dimensional unitary representation of a compact quantum group and denote by the isotropy subgroup of a linear subspace regarded as a point in the Grassmannian . We show that the space of those for which acts trivially on (or ) is open in the Zariski topology of the Weil restriction . More generally, this holds for the space of for which (a) the -action factors through its abelianization, or (b) the summands of the -representation on (or ) 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 , and can be put to similar use in fully non-commutative versions of those results (quantum graphs, quantum groups).
Paper Structure (2 sections, 12 theorems, 26 equations)

This paper contains 2 sections, 12 theorems, 26 equations.

Table of Contents

  1. Generic trivial (quantum) actions
  2. Generating operator algebras

Key Result

Theorem 1

Let $V$ be a finite-dimensional representation of a compact quantum group $\mathop{\mathrm{G}}\nolimits$, and denote by $\mathop{\mathrm{G}}\nolimits_W\le \mathop{\mathrm{G}}\nolimits$ the isotropy quantum subgroup of $W\in {\mathbb G}(d,V)$. The subsets are open in the Zariski topology on $\mathop{\mathrm{Res}}\nolimits_{{\mathbb C}/{\mathbb R}}{\mathbb G}(d,\dim V)$, and hence dense in the stan

Theorems & Definitions (24)

  • Theorem 1
  • Proposition 2
  • Theorem 3: \ref{['th:gen.zar.den.sep']}
  • Proposition 1.1
  • Proof 1
  • Proposition 1.2
  • Proof 2
  • Lemma 1.3
  • Proof 3
  • Example 1.4
  • ...and 14 more