Semi-simple element

of a linear algebraic group

An element , where is a finite-dimensional vector space over an algebraically closed field , which is a semi-simple endomorphism of the space , i.e. is diagonalizable. The notion of a semi-simple element of is intrinsic, i.e. is determined by the algebraic group structure of only and does not depend on the choice of a faithful representation as a closed algebraic subgroup of a general linear group. An element is semi-simple if and only if the right translation operator in is diagonalizable. For any rational linear representation , the set of semi-simple elements of the group is mapped onto the set of semi-simple elements of the group .

Analogously one defines semi-simple elements of the algebraic Lie algebra of ; the differential of the representation maps the set of semi-simple elements of the algebra onto the set of semi-simple elements of its image.

By definition, a semi-simple element of an abstract Lie algebra is an element for which the adjoint linear transformation is a semi-simple endomorphism of the vector space . If is the Lie algebra of a reductive linear algebraic group, then is a semi-simple element of the algebra if and only if is a semi-simple endomorphism of .

References

[1]	A. Borel, "Linear algebraic groups" , Benjamin (1969)
[2]	Yu.I. Merzlyakov, "Rational groups" , Moscow (1980) (In Russian)
[3]	J.E. Humphreys, "Linear algebraic groups" , Springer (1975)

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 of an abstract Lie algebra such that ad is a semi-simple endomorphism of is sometimes called -semi-simple.

Cf. also Jordan decomposition, 2).

References

[a1]	J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) pp. §5.4
[a2]	J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) pp. LA6.14 (Translated from French)