A study on pairings

我们得到了一般范畴中配对的一些性质。我们研究了拓扑空间、有限群、量子群、多谐函数、可微流形、伪黎曼流形的配对。我们确定了一类q^3阶的平移平面，允许有一个q^3阶的共序群。对于每一个具有可靠Haar测度的Hopf **-代数，我们将其正则表示描述为有限维不可约表示的直接和。设EPSION是一个dfn空间或具有有限开拓扑的复向量空间，（,psi）是EPSILON上的黎曼定义域（^^-,^^-,psi^^-）是（,psi）的全纯包络。那么当且仅当H^1(^-， R) = 0时，上的任何实值多谐函数等于上的全纯函数的实部。对伪黎曼流形上的对偶连接（度量共轭连接）进行了几何研究，得到如下结果：当对偶连接具有常曲率时，我们可以正则地确定一个满足广义毕达哥拉定理的类平方距离函数。我们考虑主颤振总空间的有限障碍。如果光纤是紧李群，在作用的限制下，证明了在很多情况下，总空间具有有限等变CW复的等变同伦型。

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.