Puneet Varma (Editor)

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

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 S L n ( k ) , the projective general linear group P G L n ( k ) and the special orthogonal group S O n ( k ) are all semisimple algebraic groups.
  • The general linear group G L n ( k ) is not semisimple, as its radical is non-trivial (being the multiplicative group G m ).
  • Every direct sum of simple algebraic groups is semisimple.

    • References

    Semisimple algebraic group Wikipedia
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