# 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

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.