论文标题
Lie-Group价值的共同体和离散的共同体的歧管
Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology
论文作者
论文摘要
考虑一个紧凑的集团$ g $，该$ g $由相关类别的自动形态上的真实或复杂的Banach Lie Group $ u $ $ $ $，并留下中央子组$ k \ le u $不变。我们定义$ k $ prolative连续旋转的$ {} _ kz^n（g，u）的$ {g^n \ to u} $的$ k $ prolative连续旋转的cocycles，其coboundary为$ k $ valued $（n+1）$ - cocycle;这适用于可能的非亚洲$ U $，在这种情况下$ n = 1 $。我们表明$ {} _ kz^n（g，u）$是连续地图的$ c（g^n，u）的分析submanifolds $ c（g^n，u）$ to u $，并且它们是$ k $ k $ valued cocycles的纤维带来的分离。应用程序包括：（a）$ {z^n（g，u）\ subset c（g^n，u）} $是一个分析性submanifold，其轨道在$ u $ u $（n-1）$（n-1）$ - cochains属于$ u $ u $ u $（n-u $ u $ u $ u $ u $ u $ u $ u $ u $ u $的轨道下）； （b）因此，共同体空间$ h^n（g，u）$是离散的； （c）对于Unital $ c^*$ - 代数$ a $和$ b $，带有$ a $有限维度的形态$ a \ to b $的空间是一种分析歧管，附近的形态和统一的形态是$ u（b）$的统一组合的； （d）对于$ a $ a和$ b $ banach的情况也是如此，$ a $有限尺寸和半岛; （e）以及任意$ c^*$代数中紧凑型组的投射表示空间（最后回收马丁的结果）。
Consider a compact group $G$ acting on a real or complex Banach Lie group $U$, by automorphisms in the relevant category, and leaving a central subgroup $K\le U$ invariant. We define the spaces ${}_KZ^n(G,U)$ of $K$-relative continuous cocycles as those maps ${G^n\to U}$ whose coboundary is a $K$-valued $(n+1)$-cocycle; this applies to possibly non-abelian $U$, in which case $n=1$. We show that the ${}_KZ^n(G,U)$ are analytic submanifolds of the spaces $C(G^n,U)$ of continuous maps $G^n\to U$ and that they decompose as disjoint unions of fiber bundles over manifolds of $K$-valued cocycles. Applications include: (a) the fact that ${Z^n(G,U)\subset C(G^n,U)}$ is an analytic submanifold and its orbits under the adjoint of the group of $U$-valued $(n-1)$-cochains are open; (b) hence the cohomology spaces $H^n(G,U)$ are discrete; (c) for unital $C^*$-algebras $A$ and $B$ with $A$ finite-dimensional the space of morphisms $A\to B$ is an analytic manifold and nearby morphisms are conjugate under the unitary group $U(B)$; (d) the same goes for $A$ and $B$ Banach, with $A$ finite-dimensional and semisimple; (e) and for spaces of projective representations of compact groups in arbitrary $C^*$ algebras (the last recovering a result of Martin's).
