View source for Lie group, supersolvable

''triangular Lie group''

A connected real [[Lie group|Lie group]] $G$ for which the eigen values of the operators $\mathrm{Ad}\,g$ of adjoint representation (cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]) are real for any element $g$.

A connected Lie group $G$ is supersolvable if and only if its Lie algebra $\mathfrak{g}$ is supersolvable, so a number of properties of the class of supersolvable Lie groups are parallel with properties of supersolvable Lie algebras (cf. [[Lie algebra, supersolvable|Lie algebra, supersolvable]]).

The following fixed-point theorem is true for a supersolvable Lie group [[#References|[2]]]: Any supersolvable Lie subgroup $G$ of a projective group has a fixed point in every$G$-invariant closed subset of the real projective space. There are also other analogues of properties of complex solvable Lie groups. An arbitrary connected Lie group $G$ has maximal connected supersolvable Lie groups $T$, and they are all conjugate in $G$ (see [[#References|[2]]]). To study the structure of real semi-simple Lie groups, the subgroup $T$ is often used as the real analogue of a [[Borel subgroup|Borel subgroup]].

A simply-connected supersolvable Lie group can be isomorphically imbedded in the group of real upper-triangular matrices over $\mathbb{R}$ with positive diagonal elements (which is itself supersolvable).

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> E.B. Vinberg, "The Morozov–Borel theorem for real Lie groups" ''Soviet Math. Dokl.'' , '''2''' (1961) pp. 1416–1419 ''Dokl. Akad. Nauk SSSR'' , '''141''' (1961) pp. 270–273 {{MR|0142683}} {{ZBL|0112.02505}} </TD></TR>
</table>



====Comments====
In [[#References|[1]]] the phrase "trigonalizable Lie group" is used instead of supersolvable. The literal translation of the Russian expression is fully-solvable Lie group.

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[[Category:Lie theory and generalizations]]