View source for Lie group, supersolvable

''triangular Lie group''

A connected real [[Lie group|Lie group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058700/l0587001.png" /> for which the eigen values of the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058700/l0587002.png" /> of adjoint representation (cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]) are real for any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058700/l0587003.png" />.

A connected Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058700/l0587004.png" /> is supersolvable if and only if its Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058700/l0587005.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058700/l0587006.png" /> of a projective group has a fixed point in every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058700/l0587007.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058700/l0587008.png" /> has maximal connected supersolvable Lie groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058700/l0587009.png" />, and they are all conjugate in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058700/l05870010.png" /> (see [[#References|[2]]]). To study the structure of real semi-simple Lie groups, the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058700/l05870011.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058700/l05870012.png" /> 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)</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</TD></TR></table>



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