通过量子化从射影表示到五角上同调
From projective representations to pentagonal cohomology via quantization
AI总结：
针对满足特定条件的局部紧群，本研究通过量子化构造一族酉对偶$2$-上闭链，证明其关联的局部紧量子群同构于由对偶半直积子群与五角上闭链确定的上闭链双交叉积量子群。
摘要：
给定局部紧群$G=Q\ltimes V$满足$V$为阿贝尔群，且$Q$在庞特里亚金对偶$\hat V$上的作用存在一个满测度的自由轨道，我们构造了一族酉对偶$2$-上闭链$Ω_ω$（又称非形式德林费尔德扭），其等价类$[Ω_ω]\in H^2(\hat G,\mathbb T)$由上同调类$[ω]\in H^2(Q,\mathbb T)$参数化。我们证明相关的局部紧量子群同构于上闭链双交叉积量子群，后者对应于对偶半直积$Q\ltimes\hat V$的一对均同构于$Q$的子群，以及由群上闭链$ω$显式给出的五角上闭链$Θ_ω$。
英文摘要：
Given a locally compact group $G=Q\ltimes V$ such that $V$ is Abelian and such that the action of $Q$ on the Pontryagin dual $\hat V$ has a free orbit of full measure, we construct a family of unitary dual $2$-cocycles $Ω_ω$ (aka non-formal Drinfel'd twists) whose equivalence classes $[Ω_ω]\in H^2(\hat G,\mathbb T)$ are parametrized by cohomology classes $[ω]\in H^2(Q,\mathbb T)$. We prove that the associated locally compact quantum groups are isomorphic to cocycle bicrossed product quantum groups associated to a pair of subgroups of the dual semidirect product $Q\ltimes\hat V$, both isomorphic to $Q$, and to a pentagonal cocycle $Θ_ω$ explicitly given in terms of the group cocycle $ω$.
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