JONATHAN BREZI N CHAPTE R II Algebrai c P r e l i m i n a r i e s 1. Notation: Lie groups will be denoted by italic capitals and will be under - stood to be connected and simply connected. The Lie algebr a of a Lie group will always be denoted by the correspondin g unitalicized capital. Let G be a Lie group. Ad will denote the adjoint r e p r e s e n t a t i o n of G on G, and ad will denote the adjoint r e p r e s e n t a t i o n of G on G. Let jS be a connected, n o r m a l subgroup of G . Ad will denote the linea r representatio n of G on S got by restrictin g Ad(G) to act on S. The correspondin g Lie algebr a notation will be ad . By Lie algebr a we shall always mea n a r e a l Lie algebra . Let L be a Lie algebra . L will denote the space of all linea r functionals on L. Let S and G r e m a i n as above. Given g G and v G , we define Ad g and ad v from S to S by [(Ad*g)/)](x) =*([Ad s g" 1 ]x) [(adgY)/](x) =M[adg(-Y)]x) for all / ) S and all x e S. In cas e S = G , we drop the subscrip t S and S, and we writ e simply Ad and ad . Ad and ad a r e r e f e r r e d to as the coadjoint representation s of G and G on G 2. Almos t algebrai c Lie algebras : Let A be a solvable Lie algebra . The m a x i m a l nilpotent ideal in A is called the n i l - r a d i c a l of A. A is said to be almos t algebrai c if its n i l - r a d i c a l N is complemente d in A by a 4
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