非交换Borsuk-Ulam型拓扑
Noncommutative Borsuk-Ulam-type conjectures
摘要：
在紧致量子群在单位C*-代数上的自由作用的框架下，我们给出了两个结果.第一个定理指出，如果$H$是紧致量子群的C*-代数，它自由地作用在一个有单位元的C*-代数$A$上，则不存在从$A$到连接C*-代数$A*H$的等变$*$ -同态.当$A$是球面上连续函数的C ~*-代数，而函数的C ~*-代数在$\mathbb{Z}/2\mathbb{Z}$上具有对映余作用时，我们恢复了著名的Borsuk-Ulam定理.第二个猜想是不存在从$H$到连接C*-代数$A*H$的等变$*$ -同态。我们展示了如何证明的特殊情况下的猜想$A=C(SU_q(2))=H$，这相当于显示的非平凡性Pflaum的量子瞬子纤维化建立从$SU_q(2)$。
英文摘要：
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 $\mathbb{Z}/2\mathbb{Z}$, 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 $A=C(SU_q(2))=H$, which is tantamount to showing the non-trivializability of Pflaum's quantum instanton fibration built from $SU_q(2)$.
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