By P. de la Harpe

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**Example text**

Such a basis J~-basis c~ is said to be h. 3B implies that two Caftan subalgebras c~ ~ are of th~ same type if a n d only if they are conjugated by an element of U(~), if and only if they are conjugated by ~q e l ~ e n t PropoSition ~C. Then there exists _h consists to e. Let h be a Cartan subalgebra a JQ-basis in ~ , of those operators in In particular, say ~ in g = s p ( ~ , J ~ ; Co). e = (en) n ~ Z~ ' such that which are diagor~l two Cartan sub algebras 36 of O ( ~ g ~ ) . 15 by an element of the group Lemma 5C.

The c-involutive Lie algebra having a Csrtan decomposition associated to Let g ~ and h is clearly s_2(00 , C). • be a classical complex Lie algebra of finite ra~Lk operators, let _h be a Cartan subalgebra of set of non zero roots of ~ with respect to 41 ~ h. 20 in particular algebras. that 6[ behaves much as for finite dimensional classical Namely i) : The set ~ proportional to ii) Let is r e d u c e d ~ are ~,~ p and root if a n d only if J ~ Z1 q + p q + ~ . the r(~,~). n(~,e). when ~ Given runs iii) n(~,~) Let ~,~ given zero root.

Si and ~ i,j E N* , i % j~ u ~j) i i,j ~ N * , i < Jl u All roots spaces corresponding ki be Then kj g h dual _h~ a l be a e = (en) n ~ Z* be a i ~ N*, let as in proposition 4B(type - s__qO(oo , C) . 19 (12) [ H, El, j - E_j,_i] = (~i - ~j)(~) (El, j - E_j,_ i) (13) [ H, Ej,_i - Ei,_j ] = (~i + ~)(H) (Ej,_i - Ei,_j) (14) [ H, E_i,j - E_j,i ] = -(k i + kj)(H) (E_i,j - E_j,i) for all Proof : as for proposition 4B(type one). Proposition 4C. subalgebra of with H E ~ • h. Let ~ ~ = s~(~,JQ; and let For each let as in proposition b~B(tFpe one).