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.
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.
- 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.
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- 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
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