Bruhat order in the Toda system on $\mathfrak{so}(2,4)$: an example of non-split real form

TL;DR

This paper explores the structure of the symmetric Toda system on the non-split real Lie algebra fso(2,4), revealing its relation to the Bruhat order of the relative Weyl group and highlighting additional complexities compared to split forms.

Contribution

It extends the understanding of Toda systems to non-split real forms, specifically fso(2,4), and clarifies how the phase diagram relates to Bruhat order with added details on trajectory dimensions.

Findings

Phase diagram linked to Bruhat order of the relative Weyl group.

Additional information needed to determine trajectory space dimensions.

Demonstrates relation between Toda system structure and Lie algebra type.

Abstract

In our previous papers we described the structure of trajectories of the symmetric Toda system on normal real forms of various Lie algebras and showed that it was totally determined by the Hasse diagram of the Bruhat order on the corresponding Weil group. This note deals with the simplest non-split real Lie algebra, $\mathfrak{so}(2,4)$. It turns out, that the phase diagram in this case is also closely related with the Bruhat order of the relative Weyl group, but a bit more information is necessary to describe the dimensions of the trajectory spaces.