具有非泛微分的量子群和有限群的Riemannian geometry
Riemannian geometry of quantum groups and finite groups with nonuniversal differentials
摘要：
利用量子框架丛理论，我们在对偶拟三角Hopf代数（如$C_q[SU_2]$及其标准双协变微分学）上构造了非交换的黎曼流形结构.度量由切空间上的辫状李代数上的辫状Killing形式和由Maurer-Cartan形式提供的$n$ -bein提供。我们也适用于理论的有限集，特别是有限群函数代数$C[G]$与微分演算和杀害形式确定的共轭类。置换群$C[S_3]$的情况下，制定了充分的细节和一个独特的扭转自由和余扭转自由或“列维-Civita”的连接，得到了非交换的Ricci曲率基本上成比例的度量（爱因斯坦空间）。我们还构造狄拉克算子的度量背景下，包括有限群，如$S_3$。在这个过程中，我们澄清建设的连接从规范场与非通用演算的量子主丛张量积形式。
英文摘要：
We construct noncommutative `Riemannian manifold' structures on dual quasitriangular Hopf algebras such as $C_q[SU_2]$ with its standard bicovariant differential calculus, using the quantum frame bundle formalism introduced previously. The metric is provided by the braided-Killing form on the braided-Lie algebra on the tangent space and the $n$-bein by the Maurer-Cartan form. We also apply the theory to finite sets and in particular to finite group function algebras $C[G]$ with differential calculi and Killing forms determined by a conjugacy class. The case of the permutation group $C[S_3]$ is worked out in full detail and a unique torsion free and cotorsion free or `Levi-Civita' connection is obtained with noncommutative Ricci curvature essentially proportional to the metric (an Einstein space). We also construct Dirac operators in the metric background, including on finite groups such as $S_3$. In the process we clarify the construction of connections from gauge fields with nonuniversal calculi on quantum principal bundles of tensor product form.
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