Semisimple algebraic group

In mathematics, especially in the areas of abstract algebra and algebraic geometry studying linear algebraic groups, a semisimple algebraic group is a type of matrix group which behaves much like a semisimple Lie algebra or semisimple ring.

Definition

A linear algebraic group is called semisimple if and only if the (solvable) radical of the identity component is trivial.

Equivalently, a semisimple linear algebraic group has no non-trivial connected, normal, abelian subgroups.

Examples

• Over an algebraically closed field $k$, the special linear group $SL_n(k)$ is semisimple.

• Every direct sum of simple algebraic groups is semisimple.