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On the geometry of quantum spheres and hyperboloids

Giovanni Landi, Chiara Pagani

Abstract

We study two classes of quantum spheres and hyperboloids which are $*$-quantum spaces for the quantum orthogonal group $\mathcal{O}(SO_q(3))$. We construct line bundles over the quantum homogeneous space of invariant elements for the quantum subgroup $SO(2)$ of $SO_q(3)$. These are associated to the quantum principal bundle via corepresentations of $SO(2)$ and are given by finitely-generated projective modules $\mathcal{E}_n$ of rank $1$ and even degree $-2n$. The corresponding idempotents, representing classes in K-theory, are explicitly worked out. For $q$ real, we diagonalise the Casimir operator of the Hopf algebra ${\mathcal{U}_{q^{1/2}}(sl_2)}$ dual to $\mathcal{O}(SO_q(3))$.

On the geometry of quantum spheres and hyperboloids

Abstract

We study two classes of quantum spheres and hyperboloids which are -quantum spaces for the quantum orthogonal group . We construct line bundles over the quantum homogeneous space of invariant elements for the quantum subgroup of . These are associated to the quantum principal bundle via corepresentations of and are given by finitely-generated projective modules of rank and even degree . The corresponding idempotents, representing classes in K-theory, are explicitly worked out. For real, we diagonalise the Casimir operator of the Hopf algebra dual to .
Paper Structure (19 sections, 10 theorems, 184 equations)

This paper contains 19 sections, 10 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. The quantum special orthogonal groups $SO_q({N})$
  3. Real forms
  4. Quantum spaces and exterior algebras
  5. The quantum orthogonal group $SO_q({3})$
  6. The quantum determinant
  7. Two real forms of $SO_q({3})$
  8. The double covering of $SO_{q}(3)$
  9. The orthogonal $2$-sphere and hyperboloid
  10. Pre-regular multilinear forms
  11. The quantum homogeneous spaces
  12. The quantum principal $SO(2)$-bundle
  13. Two $*$-quantum homogeneous spaces of $\mathcal{O}(SO_q({3}))$
  14. Line bundles
  15. The Casimir element
  16. ...and 4 more sections

Key Result

Proposition 3.1

Let $\widehat{u}= (\widehat{u}_{j k})_{j,k=1,2,3}$ be the transpose of the matrix of cofactors, $\widehat{u}_{m a}=\mathsf{cof}(u_{ })_{am}$. Then $u \widehat{u}= D_q(u) \mathrm{I}$.

Theorems & Definitions (21)

  • Proposition 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Proposition 5.1
  • ...and 11 more