By Avramov L.L. (ed.), Tchakerian K.B. (ed.)

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Based on algebra, this classi cation can be illustrated geometrically (! 77 Roots, root system). 90 Unitary groups and algebras. 8 Cartan matrix De nition Let G be a simple Lie algebra with Cartan subalgebra H and simple root system 0 = ( 1 : : : r ). The Cartan matrix A = (Aij ) of the simple Lie algebra G is the r r matrix de ned by Aij = 2 i i j i = i_ j 1 ij r where i_ = 2 i =( i i ) is the coroot (! 83) associated to the root i . This matrix plays a fundamental role in the theory of the simple Lie algebras: in particular, it is the basic ingredient for the description of the Lie algebra 22 Lie Algebras in the so-called Serre{Chevalley basis (!

Indeed, one can check that if d and d are derivations of G , then for all X Y 2 G : h i h i (dd0 ; d0 d)( X Y ]) = (dd0 ; d0 d)(X ) Y + X (dd0 ; d0 d)(Y ) and denoting Aut(G ) the group of automorphisms of G , its Lie algebra is actually the algebra of the derivations of G which will be denoted Der G . In particular, h i adX : Y 7! adX (Y ) = X Y is a derivation of G . These derivations are called inner derivations of G . They form an ideal Inder G of Der G . The algebra Inder G can be identi ed with the algebra of the group Int(G ), which is also the algebra of the group Int(G) of inner automorphisms of G, where G is a Lie group whose Lie algebra is G.

The coe cients N satisfy for any pair of roots and N 2 = 12 k(k0 + 1) 2 where k and k0 are integers such that + k and ; k0 are roots. Moreover, one has N = ;N = ;N; ; Lie Algebras 19 where the roots are normalized such that X ij X = ij =) 6=0 6=0 2=r in accordance with the Killing form (! 44) 0 BB BB gij = B BB @ ij 0 1 1 0 0 |{z} | r 0 ... {z n;r=2p 1 C C C C C C C 1A 0 1 0 } De nition The set H generated by the generators (H1 : : : Hr ) is called the Cartan subalgebra of G . The dimension of the Cartan subalgebra is the rank of the Lie algebra G : rank G = dim H Property The Cartan subalgebra H is unique up to conjugation of G .