# 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 a field , the special linear group , the projective general linear group and the special orthogonal group are all semisimple algebraic groups.
- The general linear group is
*not*semisimple, as its radical is non-trivial (being the multiplicative group ). - Every direct sum of simple algebraic groups is semisimple.

## Properties

## References

- Borel, Armand (1991),
*Linear algebraic groups*, Graduate Texts in Mathematics,**126**(2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-97370-8, MR 1102012 - Humphreys, James E. (1972),
*Linear Algebraic Groups*, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90108-4, MR 0396773 - Springer, Tonny A. (1998),
*Linear algebraic groups*, Progress in Mathematics,**9**(2nd ed.), Boston, MA: BirkhĂ¤user Boston, ISBN 978-0-8176-4021-7, MR 1642713

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