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The dimension of a [[Cartan subgroup|Cartan subgroup]] of it (this dimension does not depend on the choice of the Cartan subgroup). Along with the rank of an algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r0774901.png" /> one considers its semi-simple rank and reductive rank, which, by definition, are equal to the rank of the algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r0774902.png" /> and the rank of the algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r0774903.png" /> respectively, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r0774904.png" /> is the radical of the algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r0774905.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r0774906.png" /> is its unipotent radical (cf. [[Radical of a group|Radical of a group]]; [[Unipotent element|Unipotent element]]). The reductive rank of an algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r0774907.png" /> is equal to the dimension of any of its maximal tori (cf. [[Maximal torus|Maximal torus]]). The reductive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r0774909.png" />-rank of a [[Linear algebraic group|linear algebraic group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749010.png" /> defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749011.png" /> (and in the case when the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749012.png" /> is reductive (cf. [[Reductive group|Reductive group]]) — simply its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749014.png" />-rank) is the dimension of a maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749015.png" />-split torus of it (this dimension does not depend on the choice of a torus; see [[Split group|Split group]]). If the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749016.png" />-rank of a reductive linear algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749017.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749018.png" /> is zero (is equal to the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749019.png" />), then the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749020.png" /> is said to be anisotropic (or split, respectively) over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749021.png" /> (see also [[Anisotropic group|Anisotropic group]]).
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 +
The dimension of a [[Cartan subgroup|Cartan subgroup]] of it (this dimension does not depend on the choice of the Cartan subgroup). Along with the rank of an algebraic group $  G $
 +
one considers its semi-simple rank and reductive rank, which, by definition, are equal to the rank of the algebraic group $  G / R $
 +
and the rank of the algebraic group $  G / R _{u} $
 +
respectively, where $  R $
 +
is the radical of the algebraic group $  G $
 +
and $  R _{u} $
 +
is its unipotent radical (cf. [[Radical of a group|Radical of a group]]; [[Unipotent element|Unipotent element]]). The reductive rank of an algebraic group $  G $
 +
is equal to the dimension of any of its maximal tori (cf. [[Maximal torus|Maximal torus]]). The reductive $  k $ -
 +
rank of a [[Linear algebraic group|linear algebraic group]] $  G $
 +
defined over a field $  k $ (
 +
and in the case when the group $  G $
 +
is reductive (cf. [[Reductive group|Reductive group]]) — simply its $  k $ -
 +
rank) is the dimension of a maximal $  k $ -
 +
split torus of it (this dimension does not depend on the choice of a torus; see [[Split group|Split group]]). If the $  k $ -
 +
rank of a reductive linear algebraic group $  G $
 +
over $  k $
 +
is zero (is equal to the rank of $  G $ ),  
 +
then the group $  G $
 +
is said to be anisotropic (or split, respectively) over $  k $ (
 +
see also [[Anisotropic group|Anisotropic group]]).
  
 
===Examples.===
 
===Examples.===
  
  
1) The rank of the algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749022.png" /> of all non-singular upper-triangular square matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749023.png" /> is equal to its reductive rank and equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749024.png" />; the semi-simple rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749025.png" /> is zero.
+
1) The rank of the algebraic group $  T _{n} $
 +
of all non-singular upper-triangular square matrices of order $  n $
 +
is equal to its reductive rank and equal to $  n $ ;  
 +
the semi-simple rank of $  T _{n} $
 +
is zero.
  
2) The rank of the algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749026.png" /> of all upper-triangular square matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749027.png" /> with 1 on the principal diagonal is equal to its dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749028.png" />, and the reductive and semi-simple ranks of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749029.png" /> are zero.
+
2) The rank of the algebraic group $  U _{n} $
 +
of all upper-triangular square matrices of order $  n $
 +
with 1 on the principal diagonal is equal to its dimension $  n ( n - 1 ) / 2 $ ,  
 +
and the reductive and semi-simple ranks of $  U _{n} $
 +
are zero.
  
3) The rank of the algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749030.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749031.png" />-automorphisms of a definite [[Quadratic form|quadratic form]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749032.png" /> on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749033.png" />-dimensional vector space over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749034.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749035.png" />, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749036.png" />-rank of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749037.png" /> is equal to the Witt index of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749038.png" />.
+
3) The rank of the algebraic group $  O _{n} ( k ,\  f \  ) $
 +
of all $  k $ -
 +
automorphisms of a definite [[Quadratic form|quadratic form]] $  f $
 +
on an $  n $ -
 +
dimensional vector space over a field $  k $
 +
is equal to $  [ n / 2 ] $ ,  
 +
and the $  k $ -
 +
rank of the group $  O _{n} ( k ,\  f \  ) $
 +
is equal to the Witt index of the form $  f $ .
 +
 
 +
 
 +
If the characteristic of the ground field is 0, then the rank of the algebraic group  $  G $
 +
coincides with the rank of its Lie algebra  $  L $ (
 +
see [[Rank of a Lie algebra|Rank of a Lie algebra]]) and is equal to the minimum multiplicity of the eigen value  $  \lambda = 1 $
 +
of all possible adjoint operators  $  \mathop{\rm Ad}\nolimits _{L} \  g $ (
 +
the minimum is taken over all  $  g \in G $ ).  
 +
An element  $  g \in G $
 +
for which this multiplicity is equal to the rank of the algebraic group  $  G $
 +
is called regular. The set of regular elements of  $  G $
 +
is open in the [[Zariski topology|Zariski topology]] on  $  G $ .
  
If the characteristic of the ground field is 0, then the rank of the algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749039.png" /> coincides with the rank of its Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749040.png" /> (see [[Rank of a Lie algebra|Rank of a Lie algebra]]) and is equal to the minimum multiplicity of the eigen value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749041.png" /> of all possible adjoint operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749042.png" /> (the minimum is taken over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749043.png" />). An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749044.png" /> for which this multiplicity is equal to the rank of the algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749045.png" /> is called regular. The set of regular elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749046.png" /> is open in the [[Zariski topology|Zariski topology]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077490/r07749047.png" />.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Chevalley,   "Théorie des groupes de Lie" , '''2–3''' , Hermann (1952–1955)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Borel,   J. Tits,   "Groupes réductifs" ''Publ. Math. IHES'' , '''27''' (1965) pp. 55–250</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Borel,   "Linear algebraic groups" , Benjamin (1969)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.E. Humphreys,   "Linear algebraic groups" , Springer (1975)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Chevalley, "Théorie des groupes de Lie" , '''2–3''' , Hermann (1952–1955) {{MR|0068552}} {{MR|0051242}} {{MR|0019623}} {{ZBL|0186.33104}} {{ZBL|0054.01303}} {{ZBL|0063.00843}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Borel, J. Tits, "Groupes réductifs" ''Publ. Math. IHES'' , '''27''' (1965) pp. 55–250 {{MR|0207712}} {{ZBL|0145.17402}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1975) {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR></table>

Latest revision as of 15:02, 17 December 2019

The dimension of a Cartan subgroup of it (this dimension does not depend on the choice of the Cartan subgroup). Along with the rank of an algebraic group $ G $ one considers its semi-simple rank and reductive rank, which, by definition, are equal to the rank of the algebraic group $ G / R $ and the rank of the algebraic group $ G / R _{u} $ respectively, where $ R $ is the radical of the algebraic group $ G $ and $ R _{u} $ is its unipotent radical (cf. Radical of a group; Unipotent element). The reductive rank of an algebraic group $ G $ is equal to the dimension of any of its maximal tori (cf. Maximal torus). The reductive $ k $ - rank of a linear algebraic group $ G $ defined over a field $ k $ ( and in the case when the group $ G $ is reductive (cf. Reductive group) — simply its $ k $ - rank) is the dimension of a maximal $ k $ - split torus of it (this dimension does not depend on the choice of a torus; see Split group). If the $ k $ - rank of a reductive linear algebraic group $ G $ over $ k $ is zero (is equal to the rank of $ G $ ), then the group $ G $ is said to be anisotropic (or split, respectively) over $ k $ ( see also Anisotropic group).

Examples.

1) The rank of the algebraic group $ T _{n} $ of all non-singular upper-triangular square matrices of order $ n $ is equal to its reductive rank and equal to $ n $ ; the semi-simple rank of $ T _{n} $ is zero.

2) The rank of the algebraic group $ U _{n} $ of all upper-triangular square matrices of order $ n $ with 1 on the principal diagonal is equal to its dimension $ n ( n - 1 ) / 2 $ , and the reductive and semi-simple ranks of $ U _{n} $ are zero.

3) The rank of the algebraic group $ O _{n} ( k ,\ f \ ) $ of all $ k $ - automorphisms of a definite quadratic form $ f $ on an $ n $ - dimensional vector space over a field $ k $ is equal to $ [ n / 2 ] $ , and the $ k $ - rank of the group $ O _{n} ( k ,\ f \ ) $ is equal to the Witt index of the form $ f $ .


If the characteristic of the ground field is 0, then the rank of the algebraic group $ G $ coincides with the rank of its Lie algebra $ L $ ( see Rank of a Lie algebra) and is equal to the minimum multiplicity of the eigen value $ \lambda = 1 $ of all possible adjoint operators $ \mathop{\rm Ad}\nolimits _{L} \ g $ ( the minimum is taken over all $ g \in G $ ). An element $ g \in G $ for which this multiplicity is equal to the rank of the algebraic group $ G $ is called regular. The set of regular elements of $ G $ is open in the Zariski topology on $ G $ .


References

[1] C. Chevalley, "Théorie des groupes de Lie" , 2–3 , Hermann (1952–1955) MR0068552 MR0051242 MR0019623 Zbl 0186.33104 Zbl 0054.01303 Zbl 0063.00843
[2] A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES , 27 (1965) pp. 55–250 MR0207712 Zbl 0145.17402
[3] A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[4] J.E. Humphreys, "Linear algebraic groups" , Springer (1975) MR0396773 Zbl 0325.20039
How to Cite This Entry:
Rank of an algebraic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rank_of_an_algebraic_group&oldid=16419
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article