# Lie group, supersolvable

triangular Lie group

A connected real Lie group $G$ for which the eigen values of the operators $\mathrm{Ad}\,g$ of adjoint representation (cf. 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).

The following fixed-point theorem is true for a supersolvable Lie group [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 [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.

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

 [1] A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201 [2] 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 MR0142683 Zbl 0112.02505