Toda lattice and toric varieties for real split semisimple Lie algebras

TL;DR

This paper explores the topology of isospectral manifolds linked to real split semisimple Lie algebras, connecting Toda lattices to toric varieties and providing cellular decompositions via Dynkin diagrams.

Contribution

It introduces a novel topological description of isospectral manifolds using colored-signed Dynkin diagrams and relates Toda lattice level sets to toric varieties within flag manifolds.

Findings

Manifold identified as a compact completion of a disconnected Cartan subgroup.

Cellular decomposition of the manifold using Dynkin diagrams.

Connection established between Toda lattices and toric varieties.

Abstract

The paper concerns the topology of an isospectral real smooth manifold for certain Jacobi element associated with real split semisimple Lie algebra. The manifold is identified as a compact, connected completion of the disconnected Cartan subgroup of the corresponding Lie group $\tilde G$ which is a disjoint union of the split Cartan subgroups associated to semisimple portions of Levi factors of all standard parabolic subgroups of $\tilde G$. The manifold is also related to the compactified level sets of a generalized Toda lattice equation defined on the semisimple Lie algebra, which is diffeomorphic to a toric variety in the flag manifold ${\tilde G}/B$ with Borel subgroup $B$ of $\tilde G$. We then give a cellular decomposition and the associated chain complex of the manifold by introducing colored-signed Dynkin diagrams which parametrize the cells in the decomposition.