The Full Symmetric Toda Flow and Intersections of Bruhat Cells

Symmetry Integrability and Geometry-Methods and Applications

article

Yuri B. Chernyakov¹  Georgy I. Sharygin¹  Alexander S. Sorin²  Dmitry V. Talalaev¹ 

[1] Institute for Theoretical and Experimental Physics;Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics;Institute for Information Transmission Problems;Lomonosov Moscow State University, Faculty of Mechanics and Mathematics;National Research Nuclear University MEPhI (Moscow Engineering Physics Institute);Dubna State University;Centre of integrable systems, P.G. Demidov Yaroslavl State University

关键词: Lie groups;    Bruhat order;    integrable systems;    Toda flow 2020 Mathematics Subject Classification 22E15;    70H06;   

DOI  :  10.3842/SIGMA.2020.115

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown