A study on pairings

得到了一般范畴中对偶的一些性质。本文研究了拓扑空间、有限群、量子群、多重调和函数、可微流形、伪黎曼流形中的对偶，确定了一类q^3阶平移平面，它允许一个q^3阶共线群.对于每个具有忠实Haar测度的Hopf **-代数，我们将它的正则表示描述为有限维不可约表示的直和。设EPSION是一个DFN-空间或具有有限开拓扑的复向量空间，（Ω，psi）是EPSILON上的Riemann域，（λ ^^-，Ω ^^-，psi^^-）是（Ω，psi）的全纯包络.则Ω上的任何实值多重调和函数等于Ω上的全纯函数的真实的部分当且仅当H^1（Ω ^^-，R）= 0.我们研究了对偶连接的几何性质（度量共轭连接）的伪黎曼流形，他得到了以下结果：当双重连接具有恒定曲率时，我们可以正则地确定一个满足广义勾股定理的类平方距离函数。我们考虑主纤维化的全空间的有限性障碍。如果纤维是紧李群，我们证明了在许多情况下，全空间具有有限等变CW复形的等变同伦型。

We obtained some properties of pairings in general categories. We studied pairings concerning topologcal spaces, finite groups, quantum groups, pluriharmonic functions, differentiable manifolds, pseudo-Riemannian manifolds.We determined a class of translation planes of order q^3 admitting a collineation group of order q^3.For every Hopf **-algebra with a faithful Haar measure, we describe its regular representation as a direct sum of finite dimensional irreducible representations.Let EPSION be a DFN-space or a complex vector space with the finite open topology, (OMEGA, psi) be a Riemann domain over EPSILON and (lambda^^-, OMEGA^^-, psi^^-) the envelope of holomorphy of (OMEGA, psi). Then any real-valued pluriharmonic function on OMEGA is equal to the real part of a holomorphic function on OMEGA if and only if H^1(OMEGA^^-, R) = 0.We studied geometly of dual connections (metric-conjugate connections) on pseudo-Riemannian manifolds and he obtained the following result : When a dual connection has constant curvature, we can canonically determine a square-distance-like function which satisfies the generalized Pythagorean theorem.We consider finiteness obstructions of the total space for a principal fibration. If the fiber is a compact Lie group, we proved that there are many cases that the total space has an equivariant homotopy type of a finite equivariant CW complex under the restriction of the action.