ALGEBRA MATHEMATICS

Notes on Lie Algebras Free Download

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(Cartan sub Lie algebra, roots, Weyl group, Dynkin diagram, . . . ) and the classification, as found by Killing and Cartan (the list of all semi simple Lie algebras consists of (1) the special- linear ones, i. e. all matrices (of any fixed dimension) with trace 0, (2) the orthogonal ones, i. e. all skew symmetric ma­ trices (of any fixed dimension), (3) the symplectic ones, i. e. all matrices M (of any fixed even dimension) that satisfy M J = – J MT with a certain non-degenerate skew symmetric matrix J, and (4) five special Lie algebras G2, F , E , E , E , of dimensions 14,52,78,133,248, the “exceptional Lie 4 6 7 s algebras” , that just somehow appear in the process). There is also a discus­sion of the compact form and other real forms of a (complex) semi simple Lie algebra, and a section on auto morphisms. The third chapter brings the theory of the finite dimensional representations of a semi simple Lie alge­ bra, with the highest or extreme weight as central notion

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