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Symmetry of some noncommutative sphere algebras

William J. Ugalde, Joseph C. Várilly

TL;DR

This work distinguishes two $q$-deformed 7-spheres by their $SU_q(2)$ symmetry content. By formulating and analyzing first-degree right coactions of $ ext{A}( ext{SU}_q(2))$ on finitely generated $*$-algebras, it proves that the Vaksman–Soibelman sphere $ ext{A}( ext{S}^{7}_{p})$ admits no such coaction for $m>1$, while the Brain–Landi quaternionic sphere $ ext{O}( ext{S}^{7}_{q})$ does admit coactions (including two one-parameter families). The quaternionic case yields a coaction-invariant subalgebra isomorphic to a noncommutative $ ext{S}^4_q$, and the lifted canonical map is bijective, confirming a quantum principal bundle structure. Collectively, these results establish that the two quantum 7-spheres are not isomorphic, illustrating distinct symmetry content and non-equivalence of quantum 7-sphere models.

Abstract

Two known $q$-deformed (or `quantum') $7$-spheres, both denoted $\mathbb{S}^7_q$ in the literature, may be distinguished by the presence or absence of symmetry under $\mathrm{SU}_q(2)$. The quaternionic version of $\mathbb{S}^7_q$ has been shown by Brain and Landi to support such a symmetry. Here we show that this is not the case for the older $\mathbb{S}^7_q$ introduced by Vaksman and Soibelman: and as a consequence, these quantum $7$-spheres are not isomorphic.

Symmetry of some noncommutative sphere algebras

TL;DR

This work distinguishes two -deformed 7-spheres by their symmetry content. By formulating and analyzing first-degree right coactions of on finitely generated -algebras, it proves that the Vaksman–Soibelman sphere admits no such coaction for , while the Brain–Landi quaternionic sphere does admit coactions (including two one-parameter families). The quaternionic case yields a coaction-invariant subalgebra isomorphic to a noncommutative , and the lifted canonical map is bijective, confirming a quantum principal bundle structure. Collectively, these results establish that the two quantum 7-spheres are not isomorphic, illustrating distinct symmetry content and non-equivalence of quantum 7-sphere models.

Abstract

Two known -deformed (or `quantum') -spheres, both denoted in the literature, may be distinguished by the presence or absence of symmetry under . The quaternionic version of has been shown by Brain and Landi to support such a symmetry. Here we show that this is not the case for the older introduced by Vaksman and Soibelman: and as a consequence, these quantum -spheres are not isomorphic.
Paper Structure (7 sections, 18 theorems, 89 equations, 2 tables)

This paper contains 7 sections, 18 theorems, 89 equations, 2 tables.

Table of Contents

  1. Introduction
  2. A few noncommutative spheres
  3. Algebra coactions of first degree
  4. Coactions on the Vaksman--Soibelman spheres
  5. Coactions on noncommutative $3$-spheres
  6. Application to the quaternionic $7$-sphere
  7. The noncommutative $4$-sphere relations

Key Result

Lemma 3.3

The commutations relations of either algebra imply that each column of the matrices $A,B,C,D,A',C',B',D'$ contains at most one nonzero entry. Moreover, the matrices $A$ and $C = I - A$ are diagonal.

Theorems & Definitions (36)

  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 26 more