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 $k$, the special linear group $SL_{n}(k)$, the projective general linear group ${\displaystyle PGL_{n}(k)}$ and the special orthogonal group ${\displaystyle SO_{n}(k)}$ are all semisimple algebraic groups.
• The general linear group ${\displaystyle GL_{n}(k)}$ is not semisimple, as its radical is non-trivial (being the multiplicative group $\mathbb {G} _{m}$).
• 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: Birkhäuser Boston, ISBN 978-0-8176-4021-7, MR 1642713