# The $K$-theory of twisted multipullback quantum odd spheres and complex projective spaces

@article{Hajac2015TheO, title={The \$K\$-theory of twisted multipullback quantum odd spheres and complex projective spaces}, author={Piotr M. Hajac and Ryszard Nest and David Pask and Aidan Sims and Bartosz Zieli'nski}, journal={Journal of Noncommutative Geometry}, year={2015} }

We find multipullback quantum odd-dimensional spheres equipped with natural $U(1)$-actions that yield the multipullback quantum complex projective spaces constructed from Toeplitz cubes as noncommutative quotients. We prove that the noncommutative line bundles associated to multipullback quantum odd spheres are pairwise stably non-isomorphic, and that the $K$-groups of multipullback quantum complex projective spaces and odd spheres coincide with their classical counterparts. We show that these…

## 18 Citations

### Associated noncommutative vector bundles over the Vaksman–Soibelman quantum complex projective spaces

- MathematicsBanach Center Publications
- 2020

By a diagonal embedding of $U(1)$ in $SU_q(m)$, we prolongate the diagonal circle action on the Vaksman-Soibelman quantum sphere $S^{2n+1}_q$ to the $SU_q(m)$-action on the prolongated bundle. Then…

### Rank-two Milnor idempotents for the multipullback quantum complex projective plane

- Mathematics
- 2017

The $K_0$-group of the C*-algebra of multipullback quantum complex projective plane is known to be $\mathbb{Z}^3$, with one generator given by the C*-algebra itself, one given by the section module…

### Pulling back noncommutative associated vector bundles and constructing quantum quaternionic projective spaces

- Mathematics
- 2015

Our main theorem is that the pullback of an associated noncommutative vector bundle induced by an equivariant map of quantum principal bundles is a noncommutative vector bundle associated via the…

### The K-theory type of quantum CW-complexes

- Mathematics
- 2020

The multipullback quantization of complex projective spaces lacks the naive quantum CW-complex structure because the quantization of an embedding of the nskeleton into the (n + 1)-skeleton does not…

### Pullbacks and nontriviality of associated noncommutative vector bundles

- MathematicsJournal of Noncommutative Geometry
- 2018

Our main theorem is that the pullback of an associated noncommutative vector bundle induced by an equivariant map of quantum principal bundles is a noncommutative vector bundle associated via the…

### Distinguished bases in the K-theory of multipullback quantum complex projective spaces.

- Mathematics
- 2020

We construct distinguished free generators of the $K_0$-group of the C*-algebra $C(\mathbb{CP}^n_\mathcal{T})$ of the multipullback quantum complex projective space. To this end, first we prove a…

### Projective Modules over Quantum Projective Line

- Mathematics
- 2016

Taking a groupoid C*-algebra approach to the study of the quantum complex projective spaces $\mathbb{P}^{n}\left( \mathcal{T}\right) $ constructed from the multipullback quantum spheres introduced by…

### The multiplicative K-theory of compact quantum spaces

- Mathematics
- 2022

. In this paper, we reﬁne the classiﬁcation of compact quantum spaces by K-theory type initiated in [8]. We do it by introducing a multiplicative K-theory functor for unital C*-algebras taking values…

### Projections over quantum homogeneous odd-dimensional spheres

- MathematicsJournal of Functional Analysis
- 2019

### Noncommutative Borsuk–Ulam-type conjectures revisited

- MathematicsJournal of Noncommutative Geometry
- 2021

Let $H$ be the C*-algebra of a non-trivial compact quantum group acting freely on a unital C*-algebra $A$. It was recently conjectured that there does not exist an equivariant $*$-homomorphism from…

## References

SHOWING 1-10 OF 33 REFERENCES

### Noncommutative bundles over the multi-pullback quantum complex projective plane

- Mathematics
- 2015

We equip the multi-pullback $C^*$-algebra $C(S^5_H)$ of a noncommutative-deformation of the 5-sphere with a free $U(1)$-action, and show that its fixed-point subalgebra is isomorphic with the…

### Quantum complex projective spaces from Toeplitz cubes

- Mathematics
- 2012

From N -tensor powers of the Toeplitz algebra, we construct a multi-pullback C*-algebra that is a noncommutative deformation of the complex projective space P.C/. Using Birkhoff’s Representation…

### The K-Theory of Heegaard-Type Quantum 3-Spheres

- Mathematics
- 2004

We use a Heegaard splitting of the topological 3-sphere as a guiding principle to construct a family of its noncommutative deformations. The main technical point is an identification of the universal…

### Projective Quantum Spaces

- Mathematics, Physics
- 1994

Associated to the standard SUq(n) R-matrices, we introduce quantum spheres S q , projective quantum spaces CP n−1 q , and quantum Grassmann manifolds Gk(C n q ). These algebras are shown to be…

### A Locally Trivial Quantum Hopf Fibration

- Mathematics
- 2001

Abstract
The irreducible *-representations of the polynomial algebra
$\mathcal{O}(S^{3}_{pq})$
of the quantum3-sphere introduced by Calow and Matthes are classified. The K-groups of its universal…

### Bounded and unbounded Fredholm modules for quantum projective spaces

- Mathematics
- 2010

We construct explicit generators of the K-theory and K-homology of the coordinate algebras of functions on the quantum projective spaces. We also sketch a construction of unbounded Fredholm modules,…

### A Locally Trivial Quantum Hopf Fibration

- Mathematics
- 2001

The irreducible ∗ -representations of the polynomial algebra O ( S 3 pq ) of the quantum 3-sphere introduced by Calow and Matthes are classiﬁed. The K -groups of its universal C ∗ -algebra are shown…

### Bundles over Quantum Sphere and Noncommutative Index Theorem

- Mathematics
- 2000

The Noncommutative Index Theorem is used to prove that the Chern numbers of quantum Hopf line bundles over the standard Podles quantum sphere equal the winding numbers of the repres- entations…

### Quantum lens spaces and principal actions on graph C*-algebras

- Mathematics
- 2002

We study certain principal actions on noncommutative C*-algebras. Our main examples are the Z_p- and T-actions on the odd-dimensional quantum spheres, yielding as fixed-point algebras quantum lens…

### Twisted $C^*$-algebras associated to finitely aligned higher-rank graphs

- MathematicsDocumenta Mathematica
- 2014

We introduce twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs and give a comprehensive treatment of their fundamental structural properties. We establish…