Algebre lineaire by Lipschitz S.

By Lipschitz S.

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See, for example, JACOBSON, pp. 112-119. Exercise l. Find the elements of the Weyl group for SU(3) and their multiplication table. VI. Structure of Simple Lie Algebras 39 VI. More on the Structure of Simple Lie Algebras In this Chapter we shall use the results on representations just obtained to learn about the algebras themselves by considering the adjoint representation. In the adjoint representation, the Lie algebra itself serves as the vector space on which the E's and H's act. l) Before studying the adjoint representation, let us first state a few properties of simple (and semi-simple) Lie algebras which may sound intuitive or obvious, but which require real mathematical proof.

Of course, this depends on our original ordering of the basis. We denote the set of simple roots by II and the set of all roots by~. , f3 E II ::::} Cl! - f3 t/: ~. To see this, suppose that to the contrary Cl! - f3 is a root. Then either Cl! - f3 or f3 - Cl! is positive. Thus either Cl! = (Cl! ) + Cl! can be written as the sum of two positive roots which is impossible for simple roots. If Cl! , (3) :::; O. This follows from Eq. 22) because f3 is a root, but f3-Cl! is not a root. Thus in Eq. 22), m = 0, so m-p :::; O.

Clearly, the diagonal elements of the matrix are all equal to two. The matrix is not necessarily symmetric, but if Aij # 0, then Aji # 0. In fact, we have shown (see the discussion preceeding Eq. 4) ) that the only possible values for the offdiagonal matrix elements are 0, ±1, ±2, and ± 3. Indeed, since the scalar product of two different simple roots is non-positive, the off-diagonal elements can be only 0, -1, -2, and - 3. We have seen that ( , ) is a scalar product on the root space. The Schwarz inequality tells us that 46 VII.

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