By Stanley I. Grossman ; traductor: Alfonso Leal Guajardo ; revisor tecnico: Francisco Paniagua Bocanegra.
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Thus there is no contribution to the trace. The trace vanishes and the Killing form is degenerate. III. Killing Form 24 Footnotes l. We follow here JACOBSON, p. 110. 2. JACOBSON, pp. 69-70. Exercise l. Define a bilinear form on SU(3) using the three-dimensional representation as follows. Let x and y be a linear combination of the matrices in Eq. 1) and define ((x, y)) = Tr xy. e. (x, y) = Tr ad xad y. It suffices to consider x and y running over some convenient basis. IV. Structure of Simple Lie Algebras 25 IV.
This dimension, which is the same as the number of simple roots, is called the rank of the algebra. 3) where [ti, i = 1, ... n are the simple roots. 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.
69-70. Exercise l. Define a bilinear form on SU(3) using the three-dimensional representation as follows. Let x and y be a linear combination of the matrices in Eq. 1) and define ((x, y)) = Tr xy. e. (x, y) = Tr ad xad y. It suffices to consider x and y running over some convenient basis. IV. Structure of Simple Lie Algebras 25 IV. The Structure of Simple Lie Algebras Our study of the Lie algebra of SU (3) revealed that the eight generators could be divided up in an illuminating fashion. Two generators, t z and y, commuted with each other.