The usage of the representative of the nilpotent orbit in the Lie algebra sp2n to obtain some adjectives of the weighted Dynkin diagram

In this scientific paper, we describe an algorithm to test if the weighted Dynkin diagram of type -Cn corresponds to one of the nilpotent orbits of sp2n, then we defined the necessary and sufficient condition on this representative that makes the diagram even. We applied this algorithm on one of the weighted Dynkin diagrams of type –C3 to prove that it is true

Usage of Distinguished Elements to Construct Algorithm to Test Simple Lie Algebras

A Lie algebra g over a field F is a vector space together with a bilinear map [ , ] satisfying [x ,x ] = 0 in addition to Jacobi identity . A Lie subalgebra B of a Lie algebra g is said to be a Cartan subalgebra if it is a nilpotent and equals its normalizer, and it is proved that semi simple Lie algebra g decomposes into weight spaces for B. In this scientific paper we present the conception of distinguished element 0 h in finite dimensional semi simple Lie algebra over a field F has characteristic 0 and we will prove that the previous decomposition g into weight spaces for B is the same to decomposition g as a direct sum of h0 ad eigen spaces. This leads us to construct algorithm to test simple Lie algebras. We programmed the previous algorithm to test simple linear Lie algebras over a numeral field by Mathematica 5.0 program where applied this algorithm on semi simple linear Lie algebra SL(3, ) to prove that it is simple.