Haantjes代数与对角化
Haantjes Algebras and Diagonalization
摘要：
我们引入Haantjes代数的概念：它由可微流形上的一族算子域的赋值组成，每个算子域都有消失的Haantjes挠。它们还需要满足合适的相容性条件。Haantjes代数自然地推广了几个已知的有趣的几何结构，产生于黎曼几何和可积系统理论。同时，正如我们将要展示的，它们在可微流形上算子对角化理论中起着至关重要的作用。假设Haantjes代数的算子是半单的和可交换的，我们将证明存在一组局部坐标，其中所有算子可以同时对角化。此外，在一般情况下，非半单的情况下，他们同时获得，在一个适当的本地图表，块对角形式。
英文摘要：
We introduce the notion of Haantjes algebra: It consists of an assignment of a family of operator fields on a differentiable manifold, each of them with vanishing Haantjes torsion. They are also required to satisfy suitable compatibility conditions. Haantjes algebras naturally generalize several known interesting geometric structures, arising in Riemannian geometry and in the theory of integrable systems. At the same time, as we will show, they play a crucial role in the theory of diagonalization of operators on differentiable manifolds. Assuming that the operators of a Haantjes algebra are semisimple and commute, we shall prove that there exists a set of local coordinates where all operators can be diagonalized simultaneously. Moreover, in the general, non-semisimple case, they acquire simultaneously, in a suitable local chart, a block-diagonal form.
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