K-Theory and Operator Algebras by B.B. Morrel, I.M. Singer

By B.B. Morrel, I.M. Singer

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This suggest the following choice of a basis of V : Xei +ej Hi = Ei,i − Ei+l,i+l , Xei −ej = Ej,i − Ei+l,j+l , i = j, = Ei+l,j − Ej+l,i , X−ei −ej = Ej,i+l − Ei,j+l , i < j. Let h be the linear span of Hi , i = 1, . . , l. 9, and conclude that o(2l) is semisimple, and h is a Cartan subalgebra in o(2l). , αl−1 = el−1 − el , αl = el−1 + el form a basis of the root system of o(2l). The corresponding Dynkin diagram is Dl . sp(2l), l ≥ 3, is a semisimple Lie algebra of type Cl Now we equip a complex vector space V of dimension 2l with a non-degenerate skew-symmetric bilinear form, and define sp(2l) as the set of linear transformations A satisfying (14).

Explicit constructions of the exceptional semisimple Lie algebras are much more involved, see [4] and references therein for details. Exercise Determine the positive roots and hence describe b, n+ and n− for o(2l+1), sp(2l), and o(2l). 33 References [1] N. Bourbaki. Groupes et alg`ebres de Lie. Masson, Paris, 1975, 1981. W. Carter. Lie algebras and root systems. W. Carter, G. G. Macdonald. Lectures on Lie groups and Lie algebras. LMS Student Texts 32, Cambridge University Press, 1995, 1–44. [3] W.

Alg`ebres de Lie semi-simple complexes. Benjamin, 1966.

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