分段平凡主余模代数的约化
Reductions of piecewise-trivial principal comodule algebras
摘要：
设$G'$是拓扑群$G$的闭子群。主$G$ -丛$X$可约化为局部平凡主$G'$ -丛$X'$当且仅当存在$X$的局部平凡化使得所有转移函数取$G'$中的值。我们证明了这个定理的一个非交换几何对应。为此，我们采用分段平凡主余代数的概念作为局部平凡紧主丛的替换。为了说明我们的定理，首先我们定义了一个新的非交换变形的$\mathbb{Z}/2\mathbb{Z}$ -主丛$S^2\rightarrow \mathbb{R}P^2$，产生一个分段平凡的主余代数。它是量子立方体的C*-代数，其每个面由Toeplitz代数给出。$\mathbb{Z}/2\mathbb{Z}$不变子代数定义了量子$\mathbb{R}P^2$的C*-代数。它是作为Toeplitz代数的三重拉回而给出的。其次，我们把这个非交换的$\mathbb{Z}/2\mathbb{Z}$ -主丛化为一个非交换的$U(1)$ -主丛，使得前者成为后者的一个约化，从而实例化了我们的定理。此外，利用K-理论的结果，我们证明了非交换丛不是平凡的。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
