Difference between revisions of "Lie group, supersolvable"
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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]]). | 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]]. | + | 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). | 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). | ||
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| − | In [[#References|[1]]] the phrase "trigonalizable | + | 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]] | [[Category:Lie theory and generalizations]] | ||
Latest revision as of 18:20, 12 December 2019
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 |
Comments
In [1] the phrase "trigonalizable Lie group" is used instead of supersolvable. The literal translation of the Russian expression is fully-solvable Lie group.
Lie group, supersolvable. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_group,_supersolvable&oldid=34501