# Download Lie Algebras and Locally Compact Groups by Irving Kaplansky PDF

By Irving Kaplansky

This quantity provides lecture notes in accordance with the author's classes on Lie algebras and the answer of Hilbert's 5th challenge. In bankruptcy 1, "Lie Algebras," the constitution conception of semi-simple Lie algebras in attribute 0 is gifted, following the tips of Killing and Cartan. bankruptcy 2, "The constitution of in the community Compact Groups," bargains with the answer of Hilbert's 5th challenge given by way of Gleason, Montgomery, and Zipplin in 1952.

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Hence the n+1 unknowns v a n ish , and in p a rtic u la r Bn = 0. T h eorem 45 . Let L be a fin ite -d im e n s io n a l Lie a lg e b ra o v e r an a lg e b r a ic a lly c lo se d fie ld . Let u be a re g u la r elem en t of L. Let H be the su b alg eb ra c o rresp o n d in g to the c h a r a c te ris tic ro o t 0 of Ru . Then H is a C arta n su b a lg e b ra of L. P r o o f. Let L =L o +L a +L 0 +... be the d ecom p osition of L r e la tiv e to R^, and note that we a re w ritin g H fo r L^. In o r d e r to p ro ve H nilp oten t it su ffic e s , by E n gel's th e o re m (T h eo rem 13), to p ro ve th at a ll rig h t m u ltip lica tio n s by e le ­ 44 m ents of H a re n ilp oten t when th ey act on H.

Potent. 2 S in ce R^ C R, R^ is Note th at bx € R C R^ and that by T h eorem 27, S^x is n il- B y T h eorem 29, applied to the im age of R^, we deduce that S S, is nilpotent and hence has tra c e a bx 0. We conclude the sectio n w ith an analogue of T h eorem 23. 28 T h eorem 3 1 . Let L be a fin ite -d im e n s io n a l L ie a lg e b ra o v e r a fie ld of c h a r a c te ris tic 0. Suppose that L ad m its a fin ite -d im e n s io n a l re p re se n ta tio n S fo r w hich the d e riv e d fo rm f is n o n -sin g u la r.

That H is a C arta n su b a lg e b ra . T h eorem 44. We p ro ve in T h eorem 45 T h eorem 4 4 is an e le m e n ta ry p re lu d e . Let A and B be n by n m a tr ic e s o v e r any fie ld . Suppose that kA + B is n ilp oten t fo r n + 1 d istin c t s c a la r s k. Then B is nilp oten t. P r o o f. On expanding (kA + B)n = 0 we get knA n + k11"1 (BAn_1 + A B A n "2 + . . + A n - 1 B) + . . + Bn = 0. This g ives us n+1 lin e a r hom ogeneous equations in n+1 "unknowns". The m a trix o f c o e ffic ie n ts is a V anderm onde m a trix and th e r e fo r e non­ sin g u la r.