By H. S Hall

ISBN-10: 1149548274

ISBN-13: 9781149548271

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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 (! 81) this matrix is also encoded in the Dynkin diagram (! 27) of the Lie algebra. The Cartan matrix satis es the following properties: Aij 2 Z Aii = 2 and Aij 0 (i 6= j ) Aij = 0 ) Aji = 0 Aij Aji 2 f0 1 2 3g (i 6= j ) det A 6= 0 (the Cartan matrix is non-degenerate) In fact, only the following possibilities occur for the simple Lie algebras: Aij Aji = 1 for all pairs i 6= j : AN , DN , E6 , E7 , E8 .

15 Compacity De nition A Lie group G of dimension n is compact if the domain of variation of its n essential parameters a1 : : : an is compact. As an example, the rotation group in N dimensions SO(N ) is compact since the domain of variation of each of its N (N ; 1)=2 parameters is the closed and bounded subset 0 2 ] of R . The Poincare group P (3 1) is not compact, but contains as a subgroup the rotation group SO(3) which is compact. Actually, SO(3) is maximal as a compact subgroup of P (3 1), any compact subgroup of P (3 1) being a Lie Algebras 31 subgroup of SO(3), up to a conjugation.

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. Finally, in the same way that Int(G) ' G=Z (G), we can write Inder G ' G =Z (G ). 23 Derivative of a Lie algebra { Nilpotent and solvable algebras De nition Let G be a Lie algebra.

### A Short Introduction to Graphical Algebra by H. S Hall

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