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