# Semi-simple element

*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 |

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Semi-simple element.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Semi-simple_element&oldid=44285