Toda lattice and toric varieties for real split semisimple Lie algebras

Details Toda lattice and toric varieties for real split semisimple Lie algebras Toda lattice and toric varieties for real split semisimple Lie algebras

1999-12-02

arxiv.org/abs/math/9912021v2

Mathematics

Symplectic Geometry

49 pages, AMSTeX, Rport no: OSU MRI-99-17, corrected some typos, added some references and two figures, sbmitted to PMJ (pmj.c

Scientific paper

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.

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