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''triangular Lie group''
 
''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" />.
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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 <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]]).
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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]]).
  
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]].
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058700/l05870012.png" /> with positive diagonal elements (which is itself supersolvable).
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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====
 
====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>
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<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>
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<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>
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</table>
  
  
  
 
====Comments====
 
====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.
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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]]

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.

How to Cite This Entry:
Lie group, supersolvable. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_group,_supersolvable&oldid=15494
This article was adapted from an original article by V.V. Gorbatsevich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article