View source for Replica of an endomorphism

''<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''

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" />.

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" />.

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" />.

====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>