In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, especially in the areas of abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

 and algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

 studying linear algebraic group

Linear algebraic group

In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices that is defined by polynomial equations...

s, a semisimple algebraic group is a type of matrix group

Matrix group

In mathematics, a matrix group is a group G consisting of invertible matrices over some field K, usually fixed in advance, with operations of matrix multiplication and inversion. More generally, one can consider n × n matrices over a commutative ring R...

 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

Radical of an algebraic group

The radical of an algebraic group is the identity component of its maximal normal solvable subgroup.- External links :*, Encyclopaedia of Mathematics...

of the identity component is trivial.

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