Noncommutative Borsuk-Ulam-type conjectures

DOI10.4064/BC106-0-1zbMATH Open1343.46064arXiv1502.05756OpenAlexW2963761364WikidataQ123272184 ScholiaQ123272184MaRDI QIDQ3460846

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Publication date: 8 January 2016

Published in: (Search for Journal in Brave)

Abstract: Within the framework of free actions of compact quantum groups on unital C*-algebras, we propose two conjectures. The first one states that, if $H$ is the C*-algebra of a compact quantum group coacting freely on a unital C*-algebra $A$, then there is no equivariant $*$-homomorphism from $A$ to the join C*-algebra $A*H$. For $A$ being the C*-algebra of continuous functions on a sphere with the antipodal coaction of the C*-algebra of funtions on $mathbbZ/2mathbbZ$, we recover the celebrated Borsuk-Ulam theorem. The second conjecture states that there is no equivariant $*$-homomorphism from $H$ to the join C*-algebra $A*H$. We show how to prove the conjecture in the special case $A=C(S{U}_q(2))=H$, which is tantamount to showing the non-trivializability of Pflaum's quantum instanton fibration built from $S{U}_q(2)$.

Full work available at URL: https://arxiv.org/abs/1502.05756

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