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A [[Lie group|Lie group]] that is nilpotent as an abstract group (cf. [[Nilpotent group|Nilpotent group]]). An Abelian Lie group is nilpotent. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l0586601.png" /> is a [[Flag|flag]] in a finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l0586602.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l0586603.png" />, then
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A [[Lie group|Lie group]] that is nilpotent as an abstract group (cf. [[Nilpotent group|Nilpotent group]]). An Abelian Lie group is nilpotent. If $  F = \{ V _{i} \} $  is a [[Flag|flag]] in a finite-dimensional vector space $  V $  over a field $  K $ , then $$
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N (F \  )  =   \{ {g \in  \mathop{\rm GL}\nolimits (V)} : {
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g v \equiv v    \mathop{\rm mod}\nolimits \  V _{i} 
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\textrm{ for  all }  v \in V _{i} ,  i \geq 1} \}
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$$ is a nilpotent algebraic group over  $  K $ ; in a basis compatible with  $  F $  its elements are represented by triangular matrices with ones on the main diagonal. If  $  F $  is a complete flag (that is, if  $  \mathop{\rm dim}\nolimits \  V _{k} = k $ ), then the matrix nilpotent Lie group  $  N ( n , k ) $  corresponding to  $  N (F \  ) $  consists of all matrices of order  $  n =  \mathop{\rm dim}\nolimits \  V $  of the form mentioned above.
  
<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/l/l058/l058660/l0586604.png" /></td> </tr></table>
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If  $  K $  is a complete normed field, then  $  N (F \  ) $  is a nilpotent Lie group over  $  K $ . Its Lie algebra is  $  \mathfrak n (F \  ) $  (see [[Lie algebra, nilpotent|Lie algebra, nilpotent]]). More generally, the Lie algebra of a Lie group  $  G $  over a field  $  K $  of characteristic 0 is nilpotent if and only if the connected component  $  G _{0} $  of the identity of  $  G $  is nilpotent. This makes it possible to carry over to nilpotent Lie groups the properties of nilpotent Lie algebras (see [[#References|[2]]], [[#References|[4]]], [[#References|[5]]]). The group version of Engel's theorem admits the following strengthening (Kolchin's theorem): If  $  G $  is a subgroup of  $  \mathop{\rm GL}\nolimits (V) $ , where  $  V $  is a finite-dimensional vector space over an arbitrary field  $  K $ , and if every  $  g \in G $  is unipotent, then there is a complete flag  $  F $  in  $  V $  such that  $  G \subset N (F \  ) $  (and  $  G $  automatically turns out to be nilpotent) (see [[#References|[3]]]).
  
is a nilpotent algebraic group over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l0586605.png" />; in a basis compatible with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l0586606.png" /> its elements are represented by triangular matrices with ones on the main diagonal. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l0586607.png" /> is a complete flag (that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l0586608.png" />), then the matrix nilpotent Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l0586609.png" /> corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866010.png" /> consists of all matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866011.png" /> of the form mentioned above.
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Nilpotent Lie groups are solvable, so the properties of solvable Lie groups carry over them, and often in a strengthened from, since every nilpotent Lie group is triangular. A connected Lie group  $  G $  is nilpotent if and only if in canonical coordinates (see [[Lie group|Lie group]]) the group operation in  $  G $  is written polynomially [[#References|[4]]]. Every simply-connected real nilpotent Lie group $  G $  is isomorphic to an algebraic group, and moreover, to an algebraic subgroup of $  N (n ,\  \mathbf R ) $ .
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866012.png" /> is a complete normed field, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866013.png" /> is a nilpotent Lie group over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866014.png" />. Its Lie algebra is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866015.png" /> (see [[Lie algebra, nilpotent|Lie algebra, nilpotent]]). More generally, the Lie algebra of a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866016.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866017.png" /> of characteristic 0 is nilpotent if and only if the connected component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866018.png" /> of the identity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866019.png" /> is nilpotent. This makes it possible to carry over to nilpotent Lie groups the properties of nilpotent Lie algebras (see [[#References|[2]]], [[#References|[4]]], [[#References|[5]]]). The group version of Engel's theorem admits the following strengthening (Kolchin's theorem): If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866020.png" /> is a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866022.png" /> is a finite-dimensional vector space over an arbitrary field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866023.png" />, and if every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866024.png" /> is unipotent, then there is a complete flag <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866025.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866026.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866027.png" /> (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866028.png" /> automatically turns out to be nilpotent) (see [[#References|[3]]]).
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A faithful representation of  $  G $  in  $  N ( n ,\  \mathbf R ) $  can be chosen so that the automorphism group $  \mathop{\rm Aut}\nolimits \  G $  can be imbedded in  $  \mathop{\rm GL}\nolimits ( n ,\  \mathbf R ) $  as the normalizer of the image of $  G $  (see [[#References|[1]]]).
  
Nilpotent Lie groups are solvable, so the properties of solvable Lie groups carry over them, and often in a strengthened from, since every nilpotent Lie group is triangular. A connected Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866029.png" /> is nilpotent if and only if in canonical coordinates (see [[Lie group|Lie group]]) the group operation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866030.png" /> is written polynomially [[#References|[4]]]. Every simply-connected real nilpotent Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866031.png" /> is isomorphic to an algebraic group, and moreover, to an algebraic subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866032.png" />.
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If $  G $  is a connected matrix real nilpotent Lie group, then it splits into the direct product of a compact Abelian Lie group and a simply-connected Lie group. A connected linear algebraic group $  G $  over a field of characteristic 0 splits into the direct product of an Abelian normal subgroup consisting of the semi-simple elements and a normal subgroup consisting of the unipotent elements [[#References|[5]]].
 
 
A faithful representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866033.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866034.png" /> can be chosen so that the automorphism group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866035.png" /> can be imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866036.png" /> as the normalizer of the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866037.png" /> (see [[#References|[1]]]).
 
 
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866038.png" /> is a connected matrix real nilpotent Lie group, then it splits into the direct product of a compact Abelian Lie group and a simply-connected Lie group. A connected linear algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l05866039.png" /> over a field of characteristic 0 splits into the direct product of an Abelian normal subgroup consisting of the semi-simple elements and a normal subgroup consisting of the unipotent elements [[#References|[5]]].
 
  
 
Nilpotent Lie groups were formerly called special Lie groups or Lie groups of rank 0. In the representation theory of semi-simple Lie groups, when studying discrete subgroups of such groups, substantial use was made of horospherical Lie groups that are nilpotent Lie groups.
 
Nilpotent Lie groups were formerly called special Lie groups or Lie groups of rank 0. In the representation theory of semi-simple Lie groups, when studying discrete subgroups of such groups, substantial use was made of horospherical Lie groups that are nilpotent Lie groups.

Latest revision as of 18:12, 12 December 2019

A Lie group that is nilpotent as an abstract group (cf. Nilpotent group). An Abelian Lie group is nilpotent. If $ F = \{ V _{i} \} $ is a flag in a finite-dimensional vector space $ V $ over a field $ K $ , then $$ N (F \ ) = \{ {g \in \mathop{\rm GL}\nolimits (V)} : { g v \equiv v \mathop{\rm mod}\nolimits \ V _{i} \textrm{ for all } v \in V _{i} , i \geq 1} \} $$ is a nilpotent algebraic group over $ K $ ; in a basis compatible with $ F $ its elements are represented by triangular matrices with ones on the main diagonal. If $ F $ is a complete flag (that is, if $ \mathop{\rm dim}\nolimits \ V _{k} = k $ ), then the matrix nilpotent Lie group $ N ( n , k ) $ corresponding to $ N (F \ ) $ consists of all matrices of order $ n = \mathop{\rm dim}\nolimits \ V $ of the form mentioned above.

If $ K $ is a complete normed field, then $ N (F \ ) $ is a nilpotent Lie group over $ K $ . Its Lie algebra is $ \mathfrak n (F \ ) $ (see Lie algebra, nilpotent). More generally, the Lie algebra of a Lie group $ G $ over a field $ K $ of characteristic 0 is nilpotent if and only if the connected component $ G _{0} $ of the identity of $ G $ is nilpotent. This makes it possible to carry over to nilpotent Lie groups the properties of nilpotent Lie algebras (see [2], [4], [5]). The group version of Engel's theorem admits the following strengthening (Kolchin's theorem): If $ G $ is a subgroup of $ \mathop{\rm GL}\nolimits (V) $ , where $ V $ is a finite-dimensional vector space over an arbitrary field $ K $ , and if every $ g \in G $ is unipotent, then there is a complete flag $ F $ in $ V $ such that $ G \subset N (F \ ) $ (and $ G $ automatically turns out to be nilpotent) (see [3]).

Nilpotent Lie groups are solvable, so the properties of solvable Lie groups carry over them, and often in a strengthened from, since every nilpotent Lie group is triangular. A connected Lie group $ G $ is nilpotent if and only if in canonical coordinates (see Lie group) the group operation in $ G $ is written polynomially [4]. Every simply-connected real nilpotent Lie group $ G $ is isomorphic to an algebraic group, and moreover, to an algebraic subgroup of $ N (n ,\ \mathbf R ) $ .

A faithful representation of $ G $ in $ N ( n ,\ \mathbf R ) $ can be chosen so that the automorphism group $ \mathop{\rm Aut}\nolimits \ G $ can be imbedded in $ \mathop{\rm GL}\nolimits ( n ,\ \mathbf R ) $ as the normalizer of the image of $ G $ (see [1]).

If $ G $ is a connected matrix real nilpotent Lie group, then it splits into the direct product of a compact Abelian Lie group and a simply-connected Lie group. A connected linear algebraic group $ G $ over a field of characteristic 0 splits into the direct product of an Abelian normal subgroup consisting of the semi-simple elements and a normal subgroup consisting of the unipotent elements [5].

Nilpotent Lie groups were formerly called special Lie groups or Lie groups of rank 0. In the representation theory of semi-simple Lie groups, when studying discrete subgroups of such groups, substantial use was made of horospherical Lie groups that are nilpotent Lie groups.

References

[1] G. Birkhoff, "Representability of Lie algebras and Lie groups by matrices" Ann. of Math. (2) , 38 (1937) pp. 526–532 MR1503351 Zbl 0016.24402 Zbl 63.0090.01
[2] N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) MR0682756 Zbl 0319.17002
[3] J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) MR0218496 Zbl 0132.27803
[4] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) MR0514561 Zbl 0451.53038
[5] C. Chevalley, "Théorie des groupes de Lie" , 3 , Hermann (1955) MR0068552 Zbl 0186.33104 Zbl 0054.01303 Zbl 0063.00843


Comments

The theory of unitary representations of nilpotent Lie groups is well understood, and goes back to the fundamental paper [a1] of A.A. Kirillov. This theory, which is usually called the "orbit method" , has extensions to the case of solvable Lie groups, although the results are not as complete as in the nilpotent case. See also [a3].

References

[a1] A.A. Kirillov, "Unitary representations of nilpotent Lie groups" Russian Math. Surveys , 17 : 4 (1962) pp. 53–104 Uspekhi Mat. Nauk , 17 : 4 (1962) pp. 57–110 MR0142001 Zbl 0106.25001
[a2] M.S. Raghunathan, "Discrete subgroups of Lie groups" , Springer (1972) MR0507234 MR0507236 Zbl 0254.22005
[a3] L. Pukanszky, "Leçons sur les représentations des groupes" , Dunod (1967) MR0217220 Zbl 0152.01201
How to Cite This Entry:
Lie group, nilpotent. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_group,_nilpotent&oldid=21889
This article was adapted from an original article by V.V. Gorbatsevich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article