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

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.