与本质双射1-上循环相关的局部紧群的量子化
Quantization of locally compact groups associated with essentially bijective $1$-cocycles
AI总结：
该研究针对带阿贝尔核的局部紧群扩张与相容本质双射1-上循环，构造对偶酉2-上循环并证明对应形变为上循环双交叉积，推广了已有有限群等相关结果，还给出了从Yang–Baxter方程解等得到局部紧量子群的途径。
摘要：
给定局部紧群的扩张0→V→G→Q→1，其中V是阿贝尔群，且存在相容的本质双射1-上循环η:Q→V̂，我们在G上定义了一个对偶酉2-上循环，并证明Ĝ的相关形变是由Q⋉V̂的两个子群构成的匹配对所定义的上循环双交叉积。我们还从匹配对的Kac上同调视角讨论了对该构造的一种解释。\n我们的框架推广了Etingof和Gelaki针对有限群的研究、Ben David与Ginosar对其的扩展，以及我们此前关于满足对偶轨道条件的局部紧群的工作。特别地，我们能从Yang–Baxter方程的每一个对合非退化集合论解，或更一般地，从每一个brace结构，得到一个局部紧量子群。\n在技术层面，关键的创新点是构造了G在L²(Q)上的不可约射影表示，以及一个Kohn–Nirenberg型的酉量子化映射L²(G)→HS(L²(Q))。
英文摘要：
Given an extension $0\to V\to G\to Q\to1$ of locally compact groups, with $V$ abelian, and a compatible essentially bijective $1$-cocycle $η\colon Q\to\hat V$, we define a dual unitary $2$-cocycle on $G$ and show that the associated deformation of $\hat G$ is a cocycle bicrossed product defined by a matched pair of subgroups of $Q\ltimes\hat V$. We also discuss an interpretation of our construction from the point of view of Kac cohomology for matched pairs. Our setup generalizes that of Etingof and Gelaki for finite groups and its extension due to Ben David and Ginosar, as well as our earlier work on locally compact groups satisfying the dual orbit condition. In particular, we get a locally compact quantum group from every involutive nondegenerate set-theoretical solution of the Yang--Baxter equation, or more generally, from every brace structure. On the technical side, the key new points are constructions of an irreducible projective representation of $G$ on $L^2(Q)$ and a unitary quantization map $L^2(G)\to{\rm HS}(L^2(Q))$ of Kohn--Nirenberg type.
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。
