Semisimple algebraic group

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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.

Contents 1 Definition 2 Examples 3 Properties 4 References

[edit] 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 connected, normal, abelian subgroups.

[edit] Examples

• Every direct sum of simple algebraic groups is semisimple.

[edit] Properties

[edit] References

• Borel, Armand (1991), Linear algebraic groups, Graduate Texts in Mathematics, 126 (2nd ed.), Berlin, New York: Springer-Verlag, MR1102012, ISBN 978-0-387-97370-8 
• Humphreys, James E. (1972), Linear Algebraic Groups, Berlin, New York: Springer-Verlag, MR0396773, ISBN 978-0-387-90108-4 
• Springer, Tonny A. (1998), Linear algebraic groups, Progress in Mathematics, 9 (2nd ed.), Boston, MA: Birkhäuser Boston, MR1642713, ISBN 978-0-8176-4021-7