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Chernyakov, Yuri B.; Sharygin, Georgy I.; Sorin, Alexander S.; Talalaev, Dmitry V.

The full symmetric Toda flow and intersections of Bruhat cells. (English) Zbl 1471.17022

SIGMA, Symmetry Integrability Geom. Methods Appl. 16, Paper 115, 8 p. (2020).
Summary: In this short note we show that the Bruhat cells in real normal forms of semisimple Lie algebras enjoy the same property as their complex analogs: for any two elements \(w, w'\) in the Weyl group \(W(\mathfrak g)\), the corresponding real Bruhat cell \(X_w\) intersects with the dual Bruhat cell \(Y_{w'}\) iff \(w \prec w'\) in the Bruhat order on \(W(\mathfrak g)\). Here \(\mathfrak g\) is a normal real form of a semisimple complex Lie algebra \(\mathfrak g_\mathbb C\). Our reasoning is based on the properties of the Toda flows rather than on the analysis of the Weyl group action and geometric considerations.

MSC:

17B20 Simple, semisimple, reductive (super)algebras
22E15 General properties and structure of real Lie groups
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics

Keywords:

Lie groups; Bruhat order; integrable systems; Toda flow
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References:

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