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.

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

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