Jan Rudnik | Searcharxiv

arXiv2015

Noncommutative bundles over the multi-pullback quantum complex projective plane

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 $C^*$-algebra of the multi-pullback quantum complex projective plane. Our main result is the stable non-triviality of the dual tautological line bundle associated to the action. We prove it by combining Chern-Galois theory with the Milnor connecting homomorphism in $K$-theory. Using the Mayer-Vietoris six-term exact sequences and the functoriality of the Künneth formula, we also compute the $K$-groups of $C(S^5_H)$.

math.KT↗

arXiv2012

The K-theory of the triple-Toeplitz deformation of the complex projective plane

We consider a family $π^i_j\colon B_i\rightarrow B_{ij}=B_{ji}$, $i,j\in \{1,2,3\}$, $i\neq j$, of $C^*$-epimorphisms assuming that it satisfies the cocycle condition. Then we show how to compute the $K$-groups of the multi-pullback $C^*$-algebra of such a family, and examplify it in the case of the triple-Toeplitz deformation of $\mathbb{C}P^2$.

math.KT↗

arXiv2010

Reductions of piecewise-trivial principal comodule algebras

Let $G'$ be a closed subgroup of a topological group $G$. A principal $G$-bundle $X$ is reducible to a locally trivial principal $G'$-bundle $X'$ if and only if there exists a local trivialisation of $X$ such that all transition functions take values in $G'$. We prove a noncommutative-geometric counterpart of this theorem. To this end, we employ the concept of a piecewise-trivial principal comodule algebra as a replacement of a locally trivial compact principal bundle. To illustrate our theorem, first we define a new noncommutative deformation of the $\mathbb{Z}/2\mathbb{Z}$-principal bundle $S^2\rightarrow \mathbb{R}P^2$ that yields a piecewise-trivial principal comodule algebra. It is the C*-algebra of a quantum cube whose each face is given by the Toeplitz algebra. The $\mathbb{Z}/2\mathbb{Z}$-invariant subalgebra defines the C*-algebra of a quantum $\mathbb{R}P^2$. It is given as a triple-pullback of Toeplitz algebras. Next, we prolongate this noncommutative $\mathbb{Z}/2\mathbb{Z}$-principal bundle to a noncommutative $U(1)$-principal bundle, so that the former becomes a reduction of the latter thus instantiating our theorem. Moreover, using K-theory results, we prove that the prolongated noncommutative bundle is not trivial.

math.QA↗