Replica of an endomorphism

$ 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)

References

[a1]	N. Bourbaki, "Groupes et algèbres de Lie" , Hermann (1975) pp. Chapts. VII-VIII