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A subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u0954101.png" /> of a [[Linear algebraic group|linear algebraic group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u0954102.png" /> consisting of unipotent elements (cf. [[Unipotent element|Unipotent element]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u0954103.png" /> is identified with its image under an isomorphic imbedding in a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u0954104.png" /> of automorphisms of a suitable finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u0954105.png" />, then a unipotent group is a subgroup contained in the set
+
{{TEX|done}}
 +
A subgroup $  U $
 +
of a [[Linear algebraic group|linear algebraic group]] $  G $
 +
consisting of unipotent elements (cf. [[Unipotent element|Unipotent element]]). If $  G $
 +
is identified with its image under an isomorphic imbedding in a group $  \mathop{\rm GL}\nolimits (V) $
 +
of automorphisms of a suitable finite-dimensional vector space $  V $ ,
 +
then a unipotent group is a subgroup contained in the set $$
 +
\{ {g \in  \mathop{\rm GL}\nolimits (V)} : {(1 - g) ^{n} = 0} \}
 +
 +
n = \mathop{\rm dim}\nolimits \  V,
 +
$$
 +
of all unipotent automorphisms of  $  V $ .  
 +
Fixing a basis in  $  V $ ,
 +
one may identify  $  \mathop{\rm GL}\nolimits (V) $
 +
with the general linear group  $  \mathop{\rm GL}\nolimits _{n} (K) $ ,  
 +
where  $  K $
 +
is an algebraically closed ground field; the linear group  $  U $
 +
is then also called a unipotent group. An example of a unipotent group is the group  $  U _{n} (K) $
 +
of all upper-triangular matrices in  $  \mathop{\rm GL}\nolimits _{n} (K) $
 +
with 1's on the main diagonal. If  $  k $
 +
is a subfield of  $  K $
 +
and  $  U $
 +
is a unipotent subgroup in $  \mathop{\rm GL}\nolimits _{n} (k) $ ,
 +
then  $  U $
 +
is conjugate over  $  k $
 +
to some subgroup of  $  U _{n} (k) $ .
 +
In particular, all elements of  $  U $
 +
have in  $  V $
 +
a common non-zero fixed vector, and  $  U $
 +
is a nilpotent group. This theorem shows that the unipotent algebraic groups are precisely the Zariski-closed subgroups of  $  U _{n} (k) $
 +
for varying  $  n $ .
  
<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/u/u095/u095410/u0954106.png" /></td> </tr></table>
 
  
of all unipotent automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u0954107.png" />. Fixing a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u0954108.png" />, one may identify <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u0954109.png" /> with the general linear group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541011.png" /> is an algebraically closed ground field; the linear group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541012.png" /> is then also called a unipotent group. An example of a unipotent group is the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541013.png" /> of all upper-triangular matrices in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541014.png" /> with 1's on the main diagonal. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541015.png" /> is a subfield of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541017.png" /> is a unipotent subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541018.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541019.png" /> is conjugate over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541020.png" /> to some subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541021.png" />. In particular, all elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541022.png" /> have in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541023.png" /> a common non-zero fixed vector, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541024.png" /> is a nilpotent group. This theorem shows that the unipotent algebraic groups are precisely the Zariski-closed subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541025.png" /> for varying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541026.png" />.
+
In any linear algebraic group  $  H $
 +
there is a unique connected normal unipotent subgroup  $  R _{u} (H) $ (
 +
the unipotent radical) with reductive quotient group $  H/R _{u} (H) $ (
 +
cf. [[Reductive group|Reductive group]]). To some extent this reduces the study of the structure of arbitrary groups to a study of the structure of reductive and unipotent groups. In contrast to the reductive case, the classification of unipotent algebraic groups is at present (1992) unknown.
  
In any linear algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541027.png" /> there is a unique connected normal unipotent subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541028.png" /> (the unipotent radical) with reductive quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541029.png" /> (cf. [[Reductive group|Reductive group]]). To some extent this reduces the study of the structure of arbitrary groups to a study of the structure of reductive and unipotent groups. In contrast to the reductive case, the classification of unipotent algebraic groups is at present (1992) unknown.
+
Every subgroup and quotient group of a unipotent algebraic group is again unipotent. If  $  \mathop{\rm char}\nolimits \  K = 0 $ ,
 +
then  $  U $
 +
is always connected; moreover, the exponential mapping  $  \mathop{\rm exp}\nolimits : \  \mathfrak u \rightarrow U $ (
 +
where  $  \mathfrak u $
 +
is the Lie algebra of  $  U $ )
 +
is an isomorphism of algebraic varieties; if  $  \mathop{\rm char}\nolimits \  K = p > 0 $ ,
 +
then there exist non-connected unipotent algebraic groups: e.g. the additive group  $  \mathbf G _{a} $
 +
of the ground field (which may be identified with  $  U _{2} (K) $ )
 +
is a  $  p $ -
 +
group and so contains a finite unipotent group. In a connected unipotent group $  U $
 +
there is a sequence of normal subgroups  $  U = U _{1} \supset \dots \supset U _{s} = \{ e \} $
 +
such that all quotients  $  U _{i} /U _ {i + 1} $
 +
are one-dimensional. Every connected one-dimensional unipotent algebraic group is isomorphic to  $  \mathbf G _{a} $ .  
 +
This reduces the study of connected unipotent algebraic groups to a description of iterated extensions of groups of type  $  \mathbf G _{a} $ .
  
Every subgroup and quotient group of a unipotent algebraic group is again unipotent. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541030.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541031.png" /> is always connected; moreover, the exponential mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541032.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541033.png" /> is the Lie algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541034.png" />) is an isomorphism of algebraic varieties; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541035.png" />, then there exist non-connected unipotent algebraic groups: e.g. the additive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541036.png" /> of the ground field (which may be identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541037.png" />) is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541038.png" />-group and so contains a finite unipotent group. In a connected unipotent group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541039.png" /> there is a sequence of normal subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541040.png" /> such that all quotients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541041.png" /> are one-dimensional. Every connected one-dimensional unipotent algebraic group is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541042.png" />. This reduces the study of connected unipotent algebraic groups to a description of iterated extensions of groups of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541043.png" />.
 
  
Much more is known about commutative unipotent algebraic groups (cf. [[#References|[4]]]) than in the general case. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541044.png" />, then they are precisely the algebraic groups isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541045.png" />; here, the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541046.png" /> is given by the exponential mapping. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541047.png" />, then the connected commutative unipotent algebraic groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541048.png" /> are precisely the connected commutative algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541049.png" />-groups. Now <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541050.png" /> need not be isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541051.png" />: for this it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541052.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541053.png" />. In the general case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541054.png" /> is isogenous (cf. [[Isogeny|Isogeny]]) to a product of certain special groups (so-called Witt groups, cf. [[#References|[2]]]).
+
Much more is known about commutative unipotent algebraic groups (cf. [[#References|[4]]]) than in the general case. If $  \mathop{\rm char}\nolimits \  K = 0 $ ,  
 +
then they are precisely the algebraic groups isomorphic to $  \mathbf G _{a} \times \dots \times \mathbf G _{a} $ ;  
 +
here, the isomorphism $  \mathbf G _{a} \times \dots \times \mathbf G _{a} \rightarrow U $
 +
is given by the exponential mapping. If $  \mathop{\rm char}\nolimits \  K = p > 0 $ ,  
 +
then the connected commutative unipotent algebraic groups $  U $
 +
are precisely the connected commutative algebraic $  p $ -
 +
groups. Now $  U $
 +
need not be isomorphic to $  \mathbf G _{a} \times \dots \times \mathbf G _{a} $ :  
 +
for this it is necessary and sufficient that $  g ^{p} = e $
 +
for all $  g \in U $ .  
 +
In the general case $  U $
 +
is isogenous (cf. [[Isogeny|Isogeny]]) to a product of certain special groups (so-called Witt groups, cf. [[#References|[2]]]).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541056.png" /> are connected unipotent algebraic groups and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541057.png" />, then the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541058.png" /> is isomorphic to an affine space. Any orbit of a unipotent algebraic group of automorphisms of an affine algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541059.png" /> is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u09541060.png" /> [[#References|[5]]].
+
If $  H $
 +
and $  U $
 +
are connected unipotent algebraic groups and $  H \subset U $ ,  
 +
then the variety $  U/H $
 +
is isomorphic to an affine space. Any orbit of a unipotent algebraic group of automorphisms of an affine algebraic variety $  X $
 +
is closed in $  X $ [[#References|[5]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Springer (1991) {{MR|1102012}} {{ZBL|0726.20030}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) {{MR|0103191}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1981) {{MR|0610979}} {{MR|0396773}} {{ZBL|0471.20029}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> T. Kambayachi, M. Miyanishi, M. Takeuchi, "Unipotent algebraic groups" , Springer (1974) {{MR|376696}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> R. Steinberg, "Conjugacy classes in algebraic groups" , ''Lect. notes in math.'' , '''366''' , Springer (1974) {{MR|0352279}} {{ZBL|0281.20037}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Springer (1991) {{MR|1102012}} {{ZBL|0726.20030}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) {{MR|0103191}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1981) {{MR|0610979}} {{MR|0396773}} {{ZBL|0471.20029}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> T. Kambayachi, M. Miyanishi, M. Takeuchi, "Unipotent algebraic groups" , Springer (1974) {{MR|376696}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> R. Steinberg, "Conjugacy classes in algebraic groups" , ''Lect. notes in math.'' , '''366''' , Springer (1974) {{MR|0352279}} {{ZBL|0281.20037}} </TD></TR></table>

Latest revision as of 17:51, 17 December 2019

A subgroup $ U $ of a linear algebraic group $ G $ consisting of unipotent elements (cf. Unipotent element). If $ G $ is identified with its image under an isomorphic imbedding in a group $ \mathop{\rm GL}\nolimits (V) $ of automorphisms of a suitable finite-dimensional vector space $ V $ , then a unipotent group is a subgroup contained in the set $$ \{ {g \in \mathop{\rm GL}\nolimits (V)} : {(1 - g) ^{n} = 0} \} , n = \mathop{\rm dim}\nolimits \ V, $$ of all unipotent automorphisms of $ V $ . Fixing a basis in $ V $ , one may identify $ \mathop{\rm GL}\nolimits (V) $ with the general linear group $ \mathop{\rm GL}\nolimits _{n} (K) $ , where $ K $ is an algebraically closed ground field; the linear group $ U $ is then also called a unipotent group. An example of a unipotent group is the group $ U _{n} (K) $ of all upper-triangular matrices in $ \mathop{\rm GL}\nolimits _{n} (K) $ with 1's on the main diagonal. If $ k $ is a subfield of $ K $ and $ U $ is a unipotent subgroup in $ \mathop{\rm GL}\nolimits _{n} (k) $ , then $ U $ is conjugate over $ k $ to some subgroup of $ U _{n} (k) $ . In particular, all elements of $ U $ have in $ V $ a common non-zero fixed vector, and $ U $ is a nilpotent group. This theorem shows that the unipotent algebraic groups are precisely the Zariski-closed subgroups of $ U _{n} (k) $ for varying $ n $ .


In any linear algebraic group $ H $ there is a unique connected normal unipotent subgroup $ R _{u} (H) $ ( the unipotent radical) with reductive quotient group $ H/R _{u} (H) $ ( cf. Reductive group). To some extent this reduces the study of the structure of arbitrary groups to a study of the structure of reductive and unipotent groups. In contrast to the reductive case, the classification of unipotent algebraic groups is at present (1992) unknown.

Every subgroup and quotient group of a unipotent algebraic group is again unipotent. If $ \mathop{\rm char}\nolimits \ K = 0 $ , then $ U $ is always connected; moreover, the exponential mapping $ \mathop{\rm exp}\nolimits : \ \mathfrak u \rightarrow U $ ( where $ \mathfrak u $ is the Lie algebra of $ U $ ) is an isomorphism of algebraic varieties; if $ \mathop{\rm char}\nolimits \ K = p > 0 $ , then there exist non-connected unipotent algebraic groups: e.g. the additive group $ \mathbf G _{a} $ of the ground field (which may be identified with $ U _{2} (K) $ ) is a $ p $ - group and so contains a finite unipotent group. In a connected unipotent group $ U $ there is a sequence of normal subgroups $ U = U _{1} \supset \dots \supset U _{s} = \{ e \} $ such that all quotients $ U _{i} /U _ {i + 1} $ are one-dimensional. Every connected one-dimensional unipotent algebraic group is isomorphic to $ \mathbf G _{a} $ . This reduces the study of connected unipotent algebraic groups to a description of iterated extensions of groups of type $ \mathbf G _{a} $ .


Much more is known about commutative unipotent algebraic groups (cf. [4]) than in the general case. If $ \mathop{\rm char}\nolimits \ K = 0 $ , then they are precisely the algebraic groups isomorphic to $ \mathbf G _{a} \times \dots \times \mathbf G _{a} $ ; here, the isomorphism $ \mathbf G _{a} \times \dots \times \mathbf G _{a} \rightarrow U $ is given by the exponential mapping. If $ \mathop{\rm char}\nolimits \ K = p > 0 $ , then the connected commutative unipotent algebraic groups $ U $ are precisely the connected commutative algebraic $ p $ - groups. Now $ U $ need not be isomorphic to $ \mathbf G _{a} \times \dots \times \mathbf G _{a} $ : for this it is necessary and sufficient that $ g ^{p} = e $ for all $ g \in U $ . In the general case $ U $ is isogenous (cf. Isogeny) to a product of certain special groups (so-called Witt groups, cf. [2]).

If $ H $ and $ U $ are connected unipotent algebraic groups and $ H \subset U $ , then the variety $ U/H $ is isomorphic to an affine space. Any orbit of a unipotent algebraic group of automorphisms of an affine algebraic variety $ X $ is closed in $ X $ [5].

References

[1] A. Borel, "Linear algebraic groups" , Springer (1991) MR1102012 Zbl 0726.20030
[2] J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) MR0103191
[3] J.E. Humphreys, "Linear algebraic groups" , Springer (1981) MR0610979 MR0396773 Zbl 0471.20029
[4] T. Kambayachi, M. Miyanishi, M. Takeuchi, "Unipotent algebraic groups" , Springer (1974) MR376696
[5] R. Steinberg, "Conjugacy classes in algebraic groups" , Lect. notes in math. , 366 , Springer (1974) MR0352279 Zbl 0281.20037
How to Cite This Entry:
Unipotent group. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Unipotent_group&oldid=44290
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article