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On characteristic classes of vector bundles over quantum spheres

Francesco D'Andrea, Giovanni Landi, Chiara Pagani

TL;DR

This paper develops a framework for understanding the ring structure of $K^0$ for quantum spaces by combining index-theoretic invariants from 1-summable Fredholm modules with a noncommutative Chern character. It proves that, under suitable hypotheses, $K_0(B)$ maps to the dual-number ring $\mathbb{Z}[t]/(t^2)$ in a way compatible with tensor products of bimodules, and applies this to Podleś spheres and a quantum 4-sphere $S^4_q$, where the Euler class of the instanton-like bundle generates the K-theory. The work provides explicit projections for vector bundles arising from corepresentations of $SU_q(2)$ and calculates their characteristic data, including the Euler/first Chern data, revealing a rich interplay between quantum group representations, Hopf-Galois extensions, and noncommutative geometry. Overall, the results illustrate how noncommutative vector bundles over quantum spheres retain a tractable, computable K-theory structure analogous to classical rational homology spheres, with concrete formulas for projections and their invariants.

Abstract

We study the quantization of spaces whose K-theory in the classical limit is the ring of dual numbers $\mathbb{Z}[t]/(t^2)$. For a compact Hausdorff space we recall necessary and sufficient conditions for this to hold. For a compact quantum space, we give sufficient conditions that guarantee there is a morphism of abelian groups $K_0 \to \mathbb{Z}[t]/(t^2)$ compatible with the tensor product of bimodules. Applications include the standard Podleś sphere $S^2_q$ and a quantum $4$-sphere $S^4_q$ coming from quantum symplectic groups. For the latter, the K-theory is generated by the Euler class of the instanton bundle. We give explicit formulas for the projections of vector bundles on $S^4_q$ associated to the principal $SU_q(2)$-bundle $S^7_q \to S^4_q$ via irreducible corepresentations of $SU_q(2)$, and compute their characteristic classes.

On characteristic classes of vector bundles over quantum spheres

TL;DR

This paper develops a framework for understanding the ring structure of for quantum spaces by combining index-theoretic invariants from 1-summable Fredholm modules with a noncommutative Chern character. It proves that, under suitable hypotheses, maps to the dual-number ring in a way compatible with tensor products of bimodules, and applies this to Podleś spheres and a quantum 4-sphere , where the Euler class of the instanton-like bundle generates the K-theory. The work provides explicit projections for vector bundles arising from corepresentations of and calculates their characteristic data, including the Euler/first Chern data, revealing a rich interplay between quantum group representations, Hopf-Galois extensions, and noncommutative geometry. Overall, the results illustrate how noncommutative vector bundles over quantum spheres retain a tractable, computable K-theory structure analogous to classical rational homology spheres, with concrete formulas for projections and their invariants.

Abstract

We study the quantization of spaces whose K-theory in the classical limit is the ring of dual numbers . For a compact Hausdorff space we recall necessary and sufficient conditions for this to hold. For a compact quantum space, we give sufficient conditions that guarantee there is a morphism of abelian groups compatible with the tensor product of bimodules. Applications include the standard Podleś sphere and a quantum -sphere coming from quantum symplectic groups. For the latter, the K-theory is generated by the Euler class of the instanton bundle. We give explicit formulas for the projections of vector bundles on associated to the principal -bundle via irreducible corepresentations of , and compute their characteristic classes.
Paper Structure (14 sections, 24 theorems, 155 equations)

This paper contains 14 sections, 24 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. The K-ring of a rational homology sphere
  3. Characteristic classes of noncommutative vector bundles
  4. 1-summable even Fredholm modules
  5. Tensor products
  6. A class of noncommutative vector bundles
  7. On a multiplicative structure of K-theory
  8. Modules associated to corepresentations
  9. Vector bundles on a quantum 4-sphere
  10. The instanton bundle
  11. The K-theory ring
  12. Projections from corepresentations of $SU_q(2)$
  13. $\mathbb{Z}$-modules
  14. Proof of Proposition \ref{['prop:sfereclassiche']}

Key Result

Proposition \ref{prop:next}

Let $X$ be a $2n$-dimensional oriented closed manifold, $n\geq 1$. Assume that $K^0(X)\cong\mathbb{Z}^2$ as an abelian group, and that there is a complex vector bundle $E\to X$ with rank $r$ and Chern number $c=\pm 1$. Then, $K^0(X)=\mathbb{Z}[t]/(t^2)$ with $t=r-[E]$.

Theorems & Definitions (50)

  • Proposition \ref{prop:next}
  • Proposition \ref{prop:undertheassumption}(i-ii)
  • Proposition \ref{prop:2q}
  • Proposition \ref{prop:2q}
  • proof
  • Proposition \ref{prop:2q}
  • proof
  • Example \ref{prop:2q}
  • Example \ref{prop:2q}
  • Proposition \ref{prop:2q}
  • ...and 40 more