Notes on Lie Algebras by Hans Samelson

By Hans Samelson

(Cartan sub Lie algebra, roots, Weyl workforce, Dynkin diagram, . . . ) and the type, as stumbled on by means of Killing and Cartan (the record of all semisimple Lie algebras contains (1) the exact- linear ones, i. e. all matrices (of any mounted measurement) with hint zero, (2) the orthogonal ones, i. e. all skewsymmetric ma­ trices (of any fastened dimension), (3) the symplectic ones, i. e. all matrices M (of any fastened even size) that fulfill M J = - J MT with a undeniable non-degenerate skewsymmetric matrix J, and (4) 5 designated Lie algebras G2, F , E , E , E , of dimensions 14,52,78,133,248, the "exceptional Lie four 6 7 s algebras" , that simply someway seem within the process). there's additionally a discus­ sion of the compact shape and different actual types of a (complex) semisimple Lie algebra, and a bit on automorphisms. The 3rd bankruptcy brings the idea of the finite dimensional representations of a semisimple Lie alge­ bra, with the top or severe weight as significant concept. The facts for the lifestyles of representations is an advert hoc model of the current commonplace evidence, yet avoids specific use of the Poincare-Birkhoff-Witt theorem. whole reducibility is proved, as traditional, with J. H. C. Whitehead's evidence (the first evidence, by means of H. Weyl, used to be analytical-topological and used the exis­ tence of a compact type of the gang in question). Then come H.

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Proof: [XY ], Z = − Y, [XZ] = Y, [ZX] = Y, α(Z)X = α(Z). 3. By Prop. A we have hα , hα = α(hα ) = 0 (since hα is of course not 0). We introduce the important elements Hα = (2/ hα , hα ) · hα , for α in ∆; they are the coroots (of g wr to h) and will play a considerable role. They span h (just like the hα ) and satisfy the relations α(Hα ) = 2 and [gα , g−α ] = Chα (= Ch−α ). More is true about the gα . P ROPOSITION C. For each α in ∆ the dimension of gα is 1, and gtα is 0 for t = 2, 3, . . , the multiples 2α, 3α, .

So κ(A, X), the trace of ad A · ad X , is 0, and κ is degenerate. (2) Suppose κ degenerate. Put g⊥ = {X : κ(X, Y ) = 0 for all Y in g}; this is the degeneracy subspace or radical of κ; it is not 0, by assumption. 5), and so [XY ] is in g⊥ , if X is. Obviously the restriction of κ to g⊥ is identically 0. Since the restriction of the Killing form to an ideal is the Killing form of the ideal, the Killing form of g⊥ is 0. Cartan’s√ first criterion then implies that g⊥ is solvable, and so g is not semisimple.

Iv) G2 . ..... .. ... ... .. . ....... ... ....... ....... ... ....... .. ... ....... . . .... . . ....... . ....... .... ............. ... ... ....... .. ...................... ....... ....... .. ....... .. ... ....... .... ............. . . . ....... .... .. ... .. .... ... ....... ... ..... ....... ....... ....... ... ... ... ....... ) Proof : Type A1 ⊕ A1 clearly corresponds to the case of a decomposable (not simple) root system of rank 2.

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