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''over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s0868301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s0868303.png" />-split group''
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{{TEX|done}}
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''over a field $  k $ ,  
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$  k $ -
 +
split group''
  
A [[Linear algebraic group|linear algebraic group]] defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s0868304.png" /> and containing a [[Borel subgroup|Borel subgroup]] that is split over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s0868305.png" />. Here a connected solvable linear algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s0868306.png" /> is called split over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s0868307.png" /> if it is defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s0868308.png" /> and has a composition series (cf. [[Composition sequence|Composition sequence]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s0868309.png" /> such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683010.png" /> are connected algebraic subgroups defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683011.png" /> and each quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683012.png" /> is isomorphic over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683013.png" /> to either a one-dimensional torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683014.png" /> or to the additive one-dimensional group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683015.png" />. In particular, an [[Algebraic torus|algebraic torus]] is split over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683016.png" /> if and only if it is defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683017.png" /> and is isomorphic over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683018.png" /> to the direct product of copies of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683019.png" />. For connected solvable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683020.png" />-split groups the [[Borel fixed-point theorem|Borel fixed-point theorem]] holds. A reductive linear algebraic group defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683021.png" /> is split over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683022.png" /> if and only if it has a maximal torus split over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683023.png" />, that is, if its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683024.png" />-rank coincides with its rank (see [[Rank of an algebraic group|Rank of an algebraic group]]; [[Reductive group|Reductive group]]). The image of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683025.png" />-split group under any rational homomorphism defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683026.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683027.png" />-split group. Every linear algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683028.png" /> defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683029.png" /> is split over an algebraic closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683030.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683031.png" /> is also reductive or solvable and connected, then it is split over some finite extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683032.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683033.png" /> is a perfect field, then a connected solvable linear algebraic group defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683034.png" /> is split over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683035.png" /> if and only if it can be reduced to triangular form over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683036.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683037.png" />, then a linear algebraic group defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683038.png" /> is split over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683039.png" /> if and only if its Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683040.png" /> is a split (or decomposable) Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683041.png" />; by definition, the latter means that the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683042.png" /> has a split Cartan subalgebra, that is, a [[Cartan subalgebra|Cartan subalgebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683043.png" /> for which all eigenvalues of every operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683045.png" />, belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683046.png" />.
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A [[Linear algebraic group|linear algebraic group]] defined over $  k $
 +
and containing a [[Borel subgroup|Borel subgroup]] that is split over $  k $ .  
 +
Here a connected solvable linear algebraic group $  B $
 +
is called split over $  k $
 +
if it is defined over $  k $
 +
and has a composition series (cf. [[Composition sequence|Composition sequence]]) $  B = B _{0} \supset B _{1} \supset \dots \supset B _{t} = \{ 1 \} $
 +
such that the $  B _{i} $
 +
are connected algebraic subgroups defined over $  k $
 +
and each quotient group $  B _{i} /B _ {i + 1} $
 +
is isomorphic over $  k $
 +
to either a one-dimensional torus $  G _{m} \cong  \mathop{\rm GL}\nolimits _{1} $
 +
or to the additive one-dimensional group $  G _{a} $ .  
 +
In particular, an [[Algebraic torus|algebraic torus]] is split over $  k $
 +
if and only if it is defined over $  k $
 +
and is isomorphic over $  k $
 +
to the direct product of copies of the group $  G _{m} $ .  
 +
For connected solvable $  k $ -
 +
split groups the [[Borel fixed-point theorem|Borel fixed-point theorem]] holds. A reductive linear algebraic group defined over $  k $
 +
is split over $  k $
 +
if and only if it has a maximal torus split over $  k $ ,  
 +
that is, if its $  k $ -
 +
rank coincides with its rank (see [[Rank of an algebraic group|Rank of an algebraic group]]; [[Reductive group|Reductive group]]). The image of a $  k $ -
 +
split group under any rational homomorphism defined over $  k $
 +
is a $  k $ -
 +
split group. Every linear algebraic group $  G $
 +
defined over a field $  k $
 +
is split over an algebraic closure of $  k $ ;  
 +
if $  G $
 +
is also reductive or solvable and connected, then it is split over some finite extension of $  k $ .  
 +
If $  k $
 +
is a perfect field, then a connected solvable linear algebraic group defined over $  k $
 +
is split over $  k $
 +
if and only if it can be reduced to triangular form over $  k $ .  
 +
If $  \mathop{\rm char}\nolimits \  k = 0 $ ,  
 +
then a linear algebraic group defined over $  k $
 +
is split over $  k $
 +
if and only if its Lie algebra $  L $
 +
is a split (or decomposable) Lie algebra over $  k $ ;  
 +
by definition, the latter means that the Lie algebra $  L $
 +
has a split Cartan subalgebra, that is, a [[Cartan subalgebra|Cartan subalgebra]] $  H \subset L $
 +
for which all eigenvalues of every operator $  \mathop{\rm ad}\nolimits _{L} \  h $ ,  
 +
$  h \in H $ ,  
 +
belong to $  k $ .
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683047.png" /> is the real Lie group of real points of a semi-simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683048.png" />-split algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683049.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683050.png" /> is the complexification of the Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683051.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683052.png" /> is called a normal real form of the complex Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683053.png" />.
 
  
There exist quasi-split groups (cf. [[Quasi-split group|Quasi-split group]]) over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683054.png" /> that are not split groups over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683055.png" />; the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683056.png" /> is an example for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086830/s08683057.png" />.
+
If  $  G _ {\mathbf R} $
 +
is the real Lie group of real points of a semi-simple  $  \mathbf R $ -
 +
split algebraic group  $  G $
 +
and if  $  G _ {\mathbf C} $
 +
is the complexification of the Lie group  $  G _ {\mathbf R} $ ,
 +
then  $  G _ {\mathbf R} $
 +
is called a normal real form of the complex Lie group  $  G _ {\mathbf C} $ .
 +
 
 +
 
 +
There exist quasi-split groups (cf. [[Quasi-split group|Quasi-split group]]) over a field $  k $
 +
that are not split groups over $  k $ ;  
 +
the group $  \mathop{\rm SO}\nolimits (3,\  1) $
 +
is an example for $  k = \mathbf R $ .
 +
 
  
 
====References====
 
====References====
 
<table><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><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Borel, J. Tits, "Groupes réductifs" ''Publ. Math. IHES'' , '''27''' (1965) pp. 55–150 {{MR|0207712}} {{ZBL|0145.17402}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> Yu.I. Merzlyakov, "Rational groups" , Moscow (1980) (In Russian) {{MR|0602700}} {{ZBL|0518.20032}} </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>
 
<table><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><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Borel, J. Tits, "Groupes réductifs" ''Publ. Math. IHES'' , '''27''' (1965) pp. 55–150 {{MR|0207712}} {{ZBL|0145.17402}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> Yu.I. Merzlyakov, "Rational groups" , Moscow (1980) (In Russian) {{MR|0602700}} {{ZBL|0518.20032}} </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 17:39, 17 December 2019

over a field $ k $ , $ k $ - split group

A linear algebraic group defined over $ k $ and containing a Borel subgroup that is split over $ k $ . Here a connected solvable linear algebraic group $ B $ is called split over $ k $ if it is defined over $ k $ and has a composition series (cf. Composition sequence) $ B = B _{0} \supset B _{1} \supset \dots \supset B _{t} = \{ 1 \} $ such that the $ B _{i} $ are connected algebraic subgroups defined over $ k $ and each quotient group $ B _{i} /B _ {i + 1} $ is isomorphic over $ k $ to either a one-dimensional torus $ G _{m} \cong \mathop{\rm GL}\nolimits _{1} $ or to the additive one-dimensional group $ G _{a} $ . In particular, an algebraic torus is split over $ k $ if and only if it is defined over $ k $ and is isomorphic over $ k $ to the direct product of copies of the group $ G _{m} $ . For connected solvable $ k $ - split groups the Borel fixed-point theorem holds. A reductive linear algebraic group defined over $ k $ is split over $ k $ if and only if it has a maximal torus split over $ k $ , that is, if its $ k $ - rank coincides with its rank (see Rank of an algebraic group; Reductive group). The image of a $ k $ - split group under any rational homomorphism defined over $ k $ is a $ k $ - split group. Every linear algebraic group $ G $ defined over a field $ k $ is split over an algebraic closure of $ k $ ; if $ G $ is also reductive or solvable and connected, then it is split over some finite extension of $ k $ . If $ k $ is a perfect field, then a connected solvable linear algebraic group defined over $ k $ is split over $ k $ if and only if it can be reduced to triangular form over $ k $ . If $ \mathop{\rm char}\nolimits \ k = 0 $ , then a linear algebraic group defined over $ k $ is split over $ k $ if and only if its Lie algebra $ L $ is a split (or decomposable) Lie algebra over $ k $ ; by definition, the latter means that the Lie algebra $ L $ has a split Cartan subalgebra, that is, a Cartan subalgebra $ H \subset L $ for which all eigenvalues of every operator $ \mathop{\rm ad}\nolimits _{L} \ h $ , $ h \in H $ , belong to $ k $ .


If $ G _ {\mathbf R} $ is the real Lie group of real points of a semi-simple $ \mathbf R $ - split algebraic group $ G $ and if $ G _ {\mathbf C} $ is the complexification of the Lie group $ G _ {\mathbf R} $ , then $ G _ {\mathbf R} $ is called a normal real form of the complex Lie group $ G _ {\mathbf C} $ .


There exist quasi-split groups (cf. Quasi-split group) over a field $ k $ that are not split groups over $ k $ ; the group $ \mathop{\rm SO}\nolimits (3,\ 1) $ is an example for $ k = \mathbf R $ .


References

[1] A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2] A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES , 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402
[3] Yu.I. Merzlyakov, "Rational groups" , Moscow (1980) (In Russian) MR0602700 Zbl 0518.20032
[4] J.E. Humphreys, "Linear algebraic groups" , Springer (1975) MR0396773 Zbl 0325.20039
How to Cite This Entry:
Split group. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Split_group&oldid=44286
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article