Algebra Lineal (8th Edition). v.Español. by Bernard Kolman, David R. Hill

By Bernard Kolman, David R. Hill

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E. transposition and complex conjugation): 28 Lie Algebras Linear groups: { The special linear group SL(n C ) (resp. SL(n R )) is the group of complex matrices M 2 GL(n C ) (resp. real matrices M 2 GL(n R )) with determinant 1. { The unitary group U (n) is the group of complex matrices M 2 GL(n C ) such that M y M = In. The special unitary group SU (n) is the subgroup of matrices of U (n) with determinant 1. { The unitary group U (p q) with signature (p q) is the group of complex matrices M 2 GL(p + q C ) such that M y Ip qM = Ip q.

Because of its uniqueness, one can say that H is \the" maximal Abelian subalgebra of G . e. for any X 2 G , one can choose H such that X 2 H). P For any element H = ri=1 i Hi 2 H, one can write h i H E = (H ) E where is a linearP functional on H, that is an element of the dual H of H, such that (H ) = ri=1 i i . Since there exists a unique (up to a multiplicative factor) non-degenerate symmetric bilinear form B (! 44) on the Cartan subalgebra H, one can associate to any functional 2 H an element H 2 H, such that B (H H ) = (H ) 20 Lie Algebras Now, one can de ne the scalar product of two roots and by = B (H H ) that is = (H ) = (H ).

The set of derivations of the Lie algebra G0 has the structure of a Lie algebra. 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 .

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