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A subalgebra of a finite-dimensional [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p0712601.png" /> over an algebraically closed field that contains a Borel subalgebra, i.e. a maximal solvable subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p0712602.png" /> (cf. also [[Lie algebra, solvable|Lie algebra, solvable]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p0712603.png" /> is a finite-dimensional Lie algebra over an arbitrary field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p0712604.png" />, then a subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p0712605.png" /> of it is also called a parabolic subalgebra if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p0712606.png" /> is a parabolic subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p0712607.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p0712608.png" /> is the algebraic closure of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p0712609.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126010.png" /> is an irreducible [[Linear algebraic group|linear algebraic group]] over a field of characteristic 0 and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126011.png" /> is its Lie algebra, then a subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126012.png" /> is a parabolic subalgebra in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126013.png" /> if and only if it coincides with the Lie algebra of some [[Parabolic subgroup|parabolic subgroup]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126014.png" />.
+
{{TEX|done}}
 +
A subalgebra of a finite-dimensional [[Lie algebra|Lie algebra]] $  \mathfrak g $
 +
over an algebraically closed field that contains a Borel subalgebra, i.e. a maximal solvable subalgebra of $  \mathfrak g $ (
 +
cf. also [[Lie algebra, solvable|Lie algebra, solvable]]). If $  \mathfrak g $
 +
is a finite-dimensional Lie algebra over an arbitrary field $  k $ ,  
 +
then a subalgebra $  \mathfrak p $
 +
of it is also called a parabolic subalgebra if $  \mathfrak p \otimes _{k} \overline{k}  $
 +
is a parabolic subalgebra of $  \mathfrak g \otimes _{k} \overline{k}  $ ,  
 +
where $  \overline{k}  $
 +
is the algebraic closure of the field $  k $ .  
 +
If $  G $
 +
is an irreducible [[Linear algebraic group|linear algebraic group]] over a field of characteristic 0 and $  \mathfrak g $
 +
is its Lie algebra, then a subalgebra $  \mathfrak p \subset \mathfrak g $
 +
is a parabolic subalgebra in $  \mathfrak g $
 +
if and only if it coincides with the Lie algebra of some [[Parabolic subgroup|parabolic subgroup]] of $  G $ .
  
Examples of parabolic subalgebras in the Lie algebra of all square matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126015.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126016.png" /> are the subalgebras of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126017.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126018.png" /> is an arbitrary set of natural numbers with sum equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126019.png" />), where the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126020.png" /> consists of all upper-triangular block-diagonal matrices with as diagonal blocks square matrices of orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126021.png" />.
 
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126022.png" /> be a reductive finite-dimensional Lie algebra (cf. [[Lie algebra, reductive|Lie algebra, reductive]]) over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126023.png" /> of characteristic 0, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126024.png" /> be a maximal diagonalizable subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126025.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126026.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126027.png" /> be the system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126028.png" />-roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126029.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126030.png" /> (cf. [[Root system|Root system]]), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126031.png" /> be a basis (a set of simple roots) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126032.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126033.png" /> be the group of elementary automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126034.png" />, i.e. the group generated by the automorphisms of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126035.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126036.png" /> is a nilpotent element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126037.png" />. Then every parabolic subalgebra of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126038.png" /> is transformed by some automorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126039.png" /> into one of the standard parabolic subalgebras of the type
+
Examples of parabolic subalgebras in the Lie algebra of all square matrices of order  $  n $
 +
over a field $  k $
 +
are the subalgebras of type  $  \mathfrak p ( \mu ) $ (
 +
$  \mu = (m _{1} \dots m _{s} ) $
 +
is an arbitrary set of natural numbers with sum equal to  $  n $ ),  
 +
where the algebra $  \mathfrak p ( \mu ) $
 +
consists of all upper-triangular block-diagonal matrices with as diagonal blocks square matrices of orders  $  m _{1} \dots m _{s} $ .
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126040.png" /></td> </tr></table>
 
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126041.png" /> is the [[Centralizer|centralizer]] of the subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126042.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126044.png" /> is the root subspace of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126045.png" /> corresponding to the root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126047.png" /> is an arbitrary subset of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126048.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126049.png" /> is the set of those roots in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126050.png" /> whose decomposition into the sum of simple roots from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126051.png" /> contains elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126052.png" /> only with non-negative coefficients. In this way the number of classes of parabolic subalgebras conjugate with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126053.png" /> turns out to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126055.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126056.png" />-rank of the semi-simple Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126057.png" />. In addition, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126058.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126059.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126060.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126061.png" /> is the minimal parabolic subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126062.png" />.
+
Let  $  \mathfrak g $
 +
be a reductive finite-dimensional Lie algebra (cf. [[Lie algebra, reductive|Lie algebra, reductive]]) over a field  $  k $
 +
of characteristic 0, let  $  \mathfrak f $
 +
be a maximal diagonalizable subalgebra of  $  \mathfrak g $
 +
over  $  k $ ,
 +
let  $  R $
 +
be the system of  $  k $ -
 +
roots of  $  \mathfrak g $
 +
relative to  $  \mathfrak f $ (
 +
cf. [[Root system|Root system]]), let  $  \Delta $
 +
be a basis (a set of simple roots) of  $  R $ ,
 +
and let  $  \mathop{\rm Aut}\nolimits _{e} \  \mathfrak g $
 +
be the group of elementary automorphisms of  $  \mathfrak g $ ,
 +
i.e. the group generated by the automorphisms of the form  $  \mathop{\rm exp}\nolimits \  \mathop{\rm ad}\nolimits \  x $ ,
 +
where  $  x $
 +
is a nilpotent element of  $  \mathfrak g $ .  
 +
Then every parabolic subalgebra of the Lie algebra  $  \mathfrak g $
 +
is transformed by some automorphism from  $  \mathop{\rm Aut}\nolimits _{e} \  \mathfrak g $
 +
into one of the standard parabolic subalgebras of the type $$
 +
\mathfrak p _ \Phi    =  \mathfrak g ^{0} + \sum _ {\alpha \in \Pi ( \Phi )} \mathfrak g
 +
^ \alpha  ,
 +
$$
 +
where  $  \mathfrak g ^{0} $
 +
is the [[Centralizer|centralizer]] of the subalgebra $  \mathfrak f $
 +
in $  \mathfrak g $ ,  
 +
$  \mathfrak g ^ \alpha  $
 +
is the root subspace of the Lie algebra $  \mathfrak g $
 +
corresponding to the root $  \alpha \in R $ ,  
 +
$  \Phi $
 +
is an arbitrary subset of the set $  \Delta $ ,  
 +
and $  \Pi ( \Phi ) $
 +
is the set of those roots in $  R $
 +
whose decomposition into the sum of simple roots from $  \Delta $
 +
contains elements of $  \Phi $
 +
only with non-negative coefficients. In this way the number of classes of parabolic subalgebras conjugate with respect to $  \mathop{\rm Aut}\nolimits _{e} \  \mathfrak g $
 +
turns out to be $  2 ^{r} $ ,  
 +
where $  r = | \Delta | $
 +
is the $  k $ -
 +
rank of the semi-simple Lie algebra $  [ \mathfrak g ,\  \mathfrak g] $ .  
 +
In addition, if $  \Phi _{1} \subseteq \Phi _{2} $ ,  
 +
then $  \mathfrak p _ {\Phi _{1}} \supseteq \mathfrak p _ {\Phi _{2}} $ .  
 +
In particular, $  \mathfrak p _ \emptyset  = \mathfrak g $ ,  
 +
and $  \mathfrak p _ \Delta  $
 +
is the minimal parabolic subalgebra of $  \mathfrak g $ .
 +
 
  
 
All non-reductive maximal subalgebras of finite-dimensional reductive Lie algebras over a field of characteristic 0 are parabolic subalgebras (see [[#References|[2]]], [[#References|[3]]], [[#References|[5]]]).
 
All non-reductive maximal subalgebras of finite-dimensional reductive Lie algebras over a field of characteristic 0 are parabolic subalgebras (see [[#References|[2]]], [[#References|[3]]], [[#References|[5]]]).

Latest revision as of 11:39, 17 December 2019

A subalgebra of a finite-dimensional Lie algebra $ \mathfrak g $ over an algebraically closed field that contains a Borel subalgebra, i.e. a maximal solvable subalgebra of $ \mathfrak g $ ( cf. also Lie algebra, solvable). If $ \mathfrak g $ is a finite-dimensional Lie algebra over an arbitrary field $ k $ , then a subalgebra $ \mathfrak p $ of it is also called a parabolic subalgebra if $ \mathfrak p \otimes _{k} \overline{k} $ is a parabolic subalgebra of $ \mathfrak g \otimes _{k} \overline{k} $ , where $ \overline{k} $ is the algebraic closure of the field $ k $ . If $ G $ is an irreducible linear algebraic group over a field of characteristic 0 and $ \mathfrak g $ is its Lie algebra, then a subalgebra $ \mathfrak p \subset \mathfrak g $ is a parabolic subalgebra in $ \mathfrak g $ if and only if it coincides with the Lie algebra of some parabolic subgroup of $ G $ .


Examples of parabolic subalgebras in the Lie algebra of all square matrices of order $ n $ over a field $ k $ are the subalgebras of type $ \mathfrak p ( \mu ) $ ( $ \mu = (m _{1} \dots m _{s} ) $ is an arbitrary set of natural numbers with sum equal to $ n $ ), where the algebra $ \mathfrak p ( \mu ) $ consists of all upper-triangular block-diagonal matrices with as diagonal blocks square matrices of orders $ m _{1} \dots m _{s} $ .


Let $ \mathfrak g $ be a reductive finite-dimensional Lie algebra (cf. Lie algebra, reductive) over a field $ k $ of characteristic 0, let $ \mathfrak f $ be a maximal diagonalizable subalgebra of $ \mathfrak g $ over $ k $ , let $ R $ be the system of $ k $ - roots of $ \mathfrak g $ relative to $ \mathfrak f $ ( cf. Root system), let $ \Delta $ be a basis (a set of simple roots) of $ R $ , and let $ \mathop{\rm Aut}\nolimits _{e} \ \mathfrak g $ be the group of elementary automorphisms of $ \mathfrak g $ , i.e. the group generated by the automorphisms of the form $ \mathop{\rm exp}\nolimits \ \mathop{\rm ad}\nolimits \ x $ , where $ x $ is a nilpotent element of $ \mathfrak g $ . Then every parabolic subalgebra of the Lie algebra $ \mathfrak g $ is transformed by some automorphism from $ \mathop{\rm Aut}\nolimits _{e} \ \mathfrak g $ into one of the standard parabolic subalgebras of the type $$ \mathfrak p _ \Phi = \mathfrak g ^{0} + \sum _ {\alpha \in \Pi ( \Phi )} \mathfrak g ^ \alpha , $$ where $ \mathfrak g ^{0} $ is the centralizer of the subalgebra $ \mathfrak f $ in $ \mathfrak g $ , $ \mathfrak g ^ \alpha $ is the root subspace of the Lie algebra $ \mathfrak g $ corresponding to the root $ \alpha \in R $ , $ \Phi $ is an arbitrary subset of the set $ \Delta $ , and $ \Pi ( \Phi ) $ is the set of those roots in $ R $ whose decomposition into the sum of simple roots from $ \Delta $ contains elements of $ \Phi $ only with non-negative coefficients. In this way the number of classes of parabolic subalgebras conjugate with respect to $ \mathop{\rm Aut}\nolimits _{e} \ \mathfrak g $ turns out to be $ 2 ^{r} $ , where $ r = | \Delta | $ is the $ k $ - rank of the semi-simple Lie algebra $ [ \mathfrak g ,\ \mathfrak g] $ . In addition, if $ \Phi _{1} \subseteq \Phi _{2} $ , then $ \mathfrak p _ {\Phi _{1}} \supseteq \mathfrak p _ {\Phi _{2}} $ . In particular, $ \mathfrak p _ \emptyset = \mathfrak g $ , and $ \mathfrak p _ \Delta $ is the minimal parabolic subalgebra of $ \mathfrak g $ .


All non-reductive maximal subalgebras of finite-dimensional reductive Lie algebras over a field of characteristic 0 are parabolic subalgebras (see [2], [3], [5]).

References

[1] N. Bourbaki, "Groupes et algèbres de Lie" , Hermann (1975) pp. Chapts. VII-VIII MR0682756 MR0573068 MR0271276 MR0240238 MR0132805 Zbl 0329.17002
[2] F.I. Karpelevich, "On non-semi-simple maximal subalgebras of semi-simple Lie algebras" Dokl. Akad. Nauk USSR , 76 (1951) pp. 775–778 (In Russian)
[3] V.V. Morozov, "Proof of the regularity theorem" Uspekhi Mat. Nauk , 11 (1956) pp. 191–194 (In Russian)
[4] G.D. Mostow, "On maximal subgroups of real Lie groups" Ann. of Math. , 74 (1961) pp. 503–517 MR0142687 Zbl 0109.02301
[5] A. Borel, J. Tits, "Eléments unipotents et sous-groupes paraboliques de groupes réductifs I" Invent. Math. , 12 (1971) pp. 95–104 MR0294349 Zbl 0238.20055
How to Cite This Entry:
Parabolic subalgebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parabolic_subalgebra&oldid=21904
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article