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''of a linear algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s0843501.png" />''
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''of a linear algebraic group $  G $ ''
  
An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s0843502.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s0843503.png" /> is a finite-dimensional vector space over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s0843504.png" />, which is a [[Semi-simple endomorphism|semi-simple endomorphism]] of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s0843505.png" />, i.e. is diagonalizable. The notion of a semi-simple element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s0843506.png" /> is intrinsic, i.e. is determined by the algebraic group structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s0843507.png" /> only and does not depend on the choice of a faithful representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s0843508.png" /> as a closed algebraic subgroup of a general linear group. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s0843509.png" /> is semi-simple if and only if the right translation operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435010.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435011.png" /> is diagonalizable. For any rational [[Linear representation|linear representation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435012.png" />, the set of semi-simple elements of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435013.png" /> is mapped onto the set of semi-simple elements of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435014.png" />.
 
  
Analogously one defines semi-simple elements of the algebraic Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435015.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435016.png" />; the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435017.png" /> of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435018.png" /> maps the set of semi-simple elements of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435019.png" /> onto the set of semi-simple elements of its image.
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An element  $  g \in G \subset  \mathop{\rm GL}\nolimits (V) $ ,
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where  $  V $
 +
is a finite-dimensional vector space over an algebraically closed field  $  K $ ,
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which is a [[Semi-simple endomorphism|semi-simple endomorphism]] of the space  $  V $ ,
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i.e. is diagonalizable. The notion of a semi-simple element of  $  G $
 +
is intrinsic, i.e. is determined by the algebraic group structure of  $  G $
 +
only and does not depend on the choice of a faithful representation  $  G \subset  \mathop{\rm GL}\nolimits (V) $
 +
as a closed algebraic subgroup of a general linear group. An element  $  g \in G $
 +
is semi-simple if and only if the right translation operator  $  \rho _{g} $
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in  $  K [G] $
 +
is diagonalizable. For any rational [[Linear representation|linear representation]]  $  \phi : \  G \rightarrow  \mathop{\rm GL}\nolimits (W) $ ,
 +
the set of semi-simple elements of the group  $  G $
 +
is mapped onto the set of semi-simple elements of the group  $  \phi (G) $ .
 +
 
 +
 
 +
Analogously one defines semi-simple elements of the algebraic Lie algebra $  \mathfrak g $
 +
of $  G $ ;  
 +
the differential $  d \phi : \  g \rightarrow \mathfrak g \mathfrak l (W) $
 +
of the representation $  \phi $
 +
maps the set of semi-simple elements of the algebra $  \mathfrak g $
 +
onto the set of semi-simple elements of its image.
 +
 
 +
By definition, a semi-simple element of an abstract Lie algebra  $  \mathfrak g $
 +
is an element  $  X \in \mathfrak g $
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for which the adjoint linear transformation  $  \mathop{\rm ad}\nolimits \  X $
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is a semi-simple endomorphism of the vector space  $  \mathfrak g $ .
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If  $  \mathfrak g \subset \mathfrak g \mathfrak l (V) $
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is the Lie algebra of a reductive linear algebraic group, then  $  X $
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is a semi-simple element of the algebra  $  \mathfrak g $
 +
if and only if  $  X $
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is a semi-simple endomorphism of  $  V $ .
  
By definition, a semi-simple element of an abstract Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435020.png" /> is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435021.png" /> for which the adjoint linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435022.png" /> is a semi-simple endomorphism of the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435023.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435024.png" /> is the Lie algebra of a reductive linear algebraic group, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435025.png" /> is a semi-simple element of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435026.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435027.png" /> is a semi-simple endomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435028.png" />.
 
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
Thus, the notions of a semi-simple element for an algebraic Lie algebra (the Lie algebra of a linear algebraic group) and for an abstract Lie algebra do not necessarily coincide. But they do so for the Lie algebras of reductive linear algebraic groups (and semi-simple Lie algebras). To avoid this confusion, an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435029.png" /> of an abstract Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435030.png" /> such that ad <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435031.png" /> is a semi-simple endomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435032.png" /> is sometimes called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084350/s08435034.png" />-semi-simple.
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Thus, the notions of a semi-simple element for an algebraic Lie algebra (the Lie algebra of a linear algebraic group) and for an abstract Lie algebra do not necessarily coincide. But they do so for the Lie algebras of reductive linear algebraic groups (and semi-simple Lie algebras). To avoid this confusion, an element $  X $
 +
of an abstract Lie algebra $  L $
 +
such that ad $  X $
 +
is a semi-simple endomorphism of $  L $
 +
is sometimes called $  \mathop{\rm ad}\nolimits $ -
 +
semi-simple.
  
 
Cf. also [[Jordan decomposition|Jordan decomposition]], 2).
 
Cf. also [[Jordan decomposition|Jordan decomposition]], 2).

Latest revision as of 17:35, 17 December 2019

of a linear algebraic group $ G $


An element $ g \in G \subset \mathop{\rm GL}\nolimits (V) $ , where $ V $ is a finite-dimensional vector space over an algebraically closed field $ K $ , which is a semi-simple endomorphism of the space $ V $ , i.e. is diagonalizable. The notion of a semi-simple element of $ G $ is intrinsic, i.e. is determined by the algebraic group structure of $ G $ only and does not depend on the choice of a faithful representation $ G \subset \mathop{\rm GL}\nolimits (V) $ as a closed algebraic subgroup of a general linear group. An element $ g \in G $ is semi-simple if and only if the right translation operator $ \rho _{g} $ in $ K [G] $ is diagonalizable. For any rational linear representation $ \phi : \ G \rightarrow \mathop{\rm GL}\nolimits (W) $ , the set of semi-simple elements of the group $ G $ is mapped onto the set of semi-simple elements of the group $ \phi (G) $ .


Analogously one defines semi-simple elements of the algebraic Lie algebra $ \mathfrak g $ of $ G $ ; the differential $ d \phi : \ g \rightarrow \mathfrak g \mathfrak l (W) $ of the representation $ \phi $ maps the set of semi-simple elements of the algebra $ \mathfrak g $ onto the set of semi-simple elements of its image.

By definition, a semi-simple element of an abstract Lie algebra $ \mathfrak g $ is an element $ X \in \mathfrak g $ for which the adjoint linear transformation $ \mathop{\rm ad}\nolimits \ X $ is a semi-simple endomorphism of the vector space $ \mathfrak g $ . If $ \mathfrak g \subset \mathfrak g \mathfrak l (V) $ is the Lie algebra of a reductive linear algebraic group, then $ X $ is a semi-simple element of the algebra $ \mathfrak g $ if and only if $ X $ is a semi-simple endomorphism of $ V $ .


References

[1] A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2] Yu.I. Merzlyakov, "Rational groups" , Moscow (1980) (In Russian) MR0602700 Zbl 0518.20032
[3] J.E. Humphreys, "Linear algebraic groups" , Springer (1975) MR0396773 Zbl 0325.20039


Comments

Thus, the notions of a semi-simple element for an algebraic Lie algebra (the Lie algebra of a linear algebraic group) and for an abstract Lie algebra do not necessarily coincide. But they do so for the Lie algebras of reductive linear algebraic groups (and semi-simple Lie algebras). To avoid this confusion, an element $ X $ of an abstract Lie algebra $ L $ such that ad $ X $ is a semi-simple endomorphism of $ L $ is sometimes called $ \mathop{\rm ad}\nolimits $ - semi-simple.

Cf. also Jordan decomposition, 2).

References

[a1] J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) pp. §5.4 MR0323842 Zbl 0254.17004
[a2] J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) pp. LA6.14 (Translated from French) MR0218496 Zbl 0132.27803
How to Cite This Entry:
Semi-simple element. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Semi-simple_element&oldid=44285
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article