$CP ^2$乘积的非阿贝尔动量多面体
Non-Abelian momentum polytopes for products of $CP ^2$
摘要：
这是两篇配套论文中的第一篇。联合的目的是研究在2-D中常见的点涡系统的高维推广。本文对复射影四维空间的副本积上的李群SU（3）作用的动量多面体进行了分类。对于2个副本，动量多面体只是一个线段，它可以以少数方式位于正Weyl室中。对于3个副本的产品，有8种不同类型的通用动量多面体，以及许多过渡多面体，所有这些都在这里分类。多面体的类型取决于辛形式在每个射影空间上的权。在第二篇论文中，我们使用辛约化技术来研究相互作用的广义点涡的可能动力学。所得结果可用于确定至多3个具有二重特征值的3 × 3埃尔米特矩阵之和的特征值所满足的不等式。
英文摘要：
This is the first of two companion papers. The joint aim is to study a generalization to higher dimension of the point vortex systems familiar in 2-D. In this paper we classify the momentum polytopes for the action of the Lie group SU(3) on products of copies of complex projective 4-space. For 2 copies, the momentum polytope is simply a line segment, which can sit in the positive Weyl chamber in a small number of ways. For a product of 3 copies there are 8 different types of generic momentum polytope, and numerous transition polytopes, all of which are classified here. The type of polytope depends on the weights of the symplectic form on each copy of projective space. In the second paper we use techniques of symplectic reduction to study the possible dynamics of interacting generalized point vortices. The results can be applied to determine the inequalities satisfied by the eigenvalues of the sum of up to three 3x3 Hermitian matrices with double eigenvalues.
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