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'' $  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|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|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====
<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====

====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>