有理曲面上的曲线与正交李代数的表示

Configurations of curves on rational surfaces and representations of orthogonal Lie algebras

摘要:

本文研究某类有理曲面上的有理曲线的configurations 与$D_{n}$型李代数（即正交李代数）的一个基本不可约表示（其最高权在正文中记作$\lambda_{n-2}$）之间的关系。我们发现该不可约表示可由对应的有理曲面上满足两组丢番图方程的（可约）有理曲线所给出，每组方程的解构成一个外尔群轨道。

Abstract:

We study the relation between certain rational surfaces and orthogonal Lie algebras (that is, $D_n$-Lie algebras). We find that a fundamental irreducible representation (whose highest weight is denoted by $\lambda_{n-2}$) is determined by finitely many rational curves on these surfaces satisfying two systems of Diophantine equations, and the solutions of each system of these equations form a Weyl group orbit.