论文标题
sobolev代数在谎言组和非交通性几何形状上
Sobolev algebras on Lie groups and noncommutative geometry
论文作者
论文摘要
我们表明，存在一个量子紧凑的度量空间，该空间是基于与下细胞laplacian $δ= - （x_1^2+\ cdots+cdots+x_m^2）$相关的每个sobolev代数的设置。 $（g，x）$的尺寸，其中$ 0 <α\ leq 1 $。我们还提供了该结果的局部紧凑型变体，并为实际的二阶次要算子提供了概括。我们还引入了一个紧凑的光谱三重（=非交通歧管）与紧凑型组上的每个下laplacian相关联。此外，我们表明其光谱维度等于$（g，x）$的局部尺寸。最后，我们证明了Conners频谱伪内，使我们能够恢复Carnot-Carthéodory距离。
We show that there exists a quantum compact metric space which underlies the setting of each Sobolev algebra associated to a subelliptic Laplacian $Δ=-(X_1^2+\cdots+X_m^2)$ on a compact connected Lie group $G$ if $p$ is large enough, more precisely under the (sharp) condition $p > \frac{d}α$ where $d$ is the local dimension of $(G,X)$ and where $0 < α\leq 1$. We also provide locally compact variants of this result and generalizations for real second order subelliptic operators. We also introduce a compact spectral triple (=noncommutative manifold) canonically associated to each subelliptic Laplacian on a compact group. In addition, we show that its spectral dimension is equal to the local dimension of $(G,X)$. Finally, we prove that the Connes spectral pseudo-metric allows us to recover the Carnot-Carathéodory distance.
