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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r0813301.png" /> of a finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r0813302.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r0813303.png" /> of characteristic 0''
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An element of the smallest algebraic Lie subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r0813304.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r0813305.png" /> (see [[Lie algebra, algebraic|Lie algebra, algebraic]]). An endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r0813306.png" /> is a replica of the endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r0813307.png" /> if and only if each tensor over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r0813308.png" /> that is annihilated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r0813309.png" /> is also annihilated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133010.png" />.
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Each replica of an endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133011.png" /> can be written as a polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133012.png" /> with coefficients from the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133013.png" /> and without absolute term. The semi-simple and nilpotent components of an endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133014.png" /> (see [[Jordan decomposition|Jordan decomposition]], 2) are replicas of it. A subalgebra of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133015.png" /> is algebraic if and only if it contains all replicas of all its elements. An endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133016.png" /> of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133017.png" /> is nilpotent if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133018.png" /> for any replica <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133019.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133020.png" />.
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'' $  X $
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of a finite-dimensional vector space  $  V $
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over a field  $  k $
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of characteristic 0''
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133021.png" /> be an algebraically closed field, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133022.png" /> be an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133023.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133024.png" /> be a semi-simple endomorphism of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133025.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133026.png" /> be an endomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133027.png" /> such that any eigenvector of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133028.png" /> corresponding to an eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133029.png" /> is also an eigenvector for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133030.png" />, but corresponding to the eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133031.png" />. An endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133032.png" /> is a replica of the endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133033.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133034.png" /> for some automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133035.png" /> of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081330/r08133036.png" />.
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An element of the smallest algebraic Lie subalgebra  $  \mathfrak{ gl  } ( V) $
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containing  $  X $(
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see [[Lie algebra, algebraic|Lie algebra, algebraic]]). An endomorphism  $  X  ^  \prime  \in \mathfrak{ gl  } ( V) $
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is a replica of the endomorphism  $  X $
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if and only if each tensor over  $  V $
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that is annihilated by  $  X $
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is also annihilated by  $  X  ^  \prime  $.
 +
 
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Each replica of an endomorphism  $  X $
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can be written as a polynomial in  $  X $
 +
with coefficients from the field  $  k $
 +
and without absolute term. The semi-simple and nilpotent components of an endomorphism  $  X $(
 +
see [[Jordan decomposition|Jordan decomposition]], 2) are replicas of it. A subalgebra of the Lie algebra  $  \mathfrak{ gl  } ( V) $
 +
is algebraic if and only if it contains all replicas of all its elements. An endomorphism  $  X $
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of a space  $  V $
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is nilpotent if and only if  $  \mathop{\rm Tr}  XX  ^  \prime  = 0 $
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for any replica  $  X  ^  \prime  $
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of  $  X $.
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Let  $  k $
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be an algebraically closed field, let $  \phi $
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be an automorphism of $  k $,  
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let $  X $
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be a semi-simple endomorphism of the space $  V $,  
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and let $  \phi ( X) $
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be an endomorphism of $  V $
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such that any eigenvector of $  X $
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corresponding to an eigenvalue $  \lambda $
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is also an eigenvector for $  \phi ( X) $,  
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but corresponding to the eigenvalue $  \phi ( \lambda ) $.  
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An endomorphism $  X  ^  \prime  \in \mathfrak{ gl  } ( V) $
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is a replica of the endomorphism $  X $
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if and only if $  X  ^  \prime  = \phi ( X) $
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for some automorphism $  \phi $
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of the field $  k $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.-P. Serre,  "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> , ''Theórie des algèbres de Lie. Topologie des groupes de Lie'' , ''Sem. S. Lie'' , '''Ie année 1954–1955''' , Secr. Math. Univ. Paris  (1955)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Chevalley,  "Théorie des groupes de Lie" , '''2''' , Hermann  (1951)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.-P. Serre,  "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> , ''Theórie des algèbres de Lie. Topologie des groupes de Lie'' , ''Sem. S. Lie'' , '''Ie année 1954–1955''' , Secr. Math. Univ. Paris  (1955)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Chevalley,  "Théorie des groupes de Lie" , '''2''' , Hermann  (1951)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,  "Groupes et algèbres de Lie" , Hermann  (1975)  pp. Chapts. VII-VIII</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,  "Groupes et algèbres de Lie" , Hermann  (1975)  pp. Chapts. VII-VIII</TD></TR></table>

Latest revision as of 08:11, 6 June 2020


$ X $ of a finite-dimensional vector space $ V $ over a field $ k $ of characteristic 0

An element of the smallest algebraic Lie subalgebra $ \mathfrak{ gl } ( V) $ containing $ X $( see Lie algebra, algebraic). An endomorphism $ X ^ \prime \in \mathfrak{ gl } ( V) $ is a replica of the endomorphism $ X $ if and only if each tensor over $ V $ that is annihilated by $ X $ is also annihilated by $ X ^ \prime $.

Each replica of an endomorphism $ X $ can be written as a polynomial in $ X $ with coefficients from the field $ k $ and without absolute term. The semi-simple and nilpotent components of an endomorphism $ X $( see Jordan decomposition, 2) are replicas of it. A subalgebra of the Lie algebra $ \mathfrak{ gl } ( V) $ is algebraic if and only if it contains all replicas of all its elements. An endomorphism $ X $ of a space $ V $ is nilpotent if and only if $ \mathop{\rm Tr} XX ^ \prime = 0 $ for any replica $ X ^ \prime $ of $ X $.

Let $ k $ be an algebraically closed field, let $ \phi $ be an automorphism of $ k $, let $ X $ be a semi-simple endomorphism of the space $ V $, and let $ \phi ( X) $ be an endomorphism of $ V $ such that any eigenvector of $ X $ corresponding to an eigenvalue $ \lambda $ is also an eigenvector for $ \phi ( X) $, but corresponding to the eigenvalue $ \phi ( \lambda ) $. An endomorphism $ X ^ \prime \in \mathfrak{ gl } ( V) $ is a replica of the endomorphism $ X $ if and only if $ X ^ \prime = \phi ( X) $ for some automorphism $ \phi $ of the field $ k $.

References

[1] J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French)
[2] , Theórie des algèbres de Lie. Topologie des groupes de Lie , Sem. S. Lie , Ie année 1954–1955 , Secr. Math. Univ. Paris (1955)
[3] C. Chevalley, "Théorie des groupes de Lie" , 2 , Hermann (1951)

Comments

References

[a1] N. Bourbaki, "Groupes et algèbres de Lie" , Hermann (1975) pp. Chapts. VII-VIII
How to Cite This Entry:
Replica of an endomorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Replica_of_an_endomorphism&oldid=48515
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article