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Encyclopedia > Semisimple

In mathematics, the term semisimple is used in a number of related ways, within different subjects. The common theme is the idea of a decomposition into 'simple' parts, that fit together in the cleanest way (by direct sum).

  • A semisimple ring or semisimple algebra is one that is semisimple as a module over itself.
  • A connected Lie group is called semisimple when its Lie algebra is; and the same for algebraic groups. Every finite dimensional representation of a semisimple Lie algebra, Lie group, or algebraic group in characteristic 0 is semisimple, i.e., completely reducible, but the converse is not true. (See reductive group.) Moreover, in characteristic p>0, semisimple Lie groups and Lie algebras have finite dimensional representations which are not semisimple. An element of a semisimple Lie group or Lie algebra is itself semisimple if its image in every finite-dimensional representation is semisimple in the sense of matrices.

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Citations: The semisimple conjugacy classes of finite groups of Lie type E 6 and E - Fleischmann, Janiszczak ... (737 words)
....[4] showed that a genus of semisimple elements of G F corresponds to a pair (J; w] where J 6= Delta is a proper subset of the vertex set Delta of the extended Dynkin diagram (up to W conjugacy) and [w] is a conjugacy class representative of NW (W J) W J.
As emerges from their work, the number of semisimple classes belonging to the genus (J; w] is equal to f(J; w] jC NW (W J) W J (w)j where f(J; w] is the number of t in a maximal....
In and [12] all generic semisimple genus numbers f T (Phi 1 ; w] have been calculated for all simply connected groups of type E 6 ; E 7 and E 8 (see also [6] for E 6) The computations were done by implementing the formulae in section 2 on a computer.
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