By Prof. Dr. Michel Lazard (auth.)

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In 1914, E. Cartan posed the matter of discovering all irreducible genuine linear Lie algebras. Iwahori gave an up to date exposition of Cartan's paintings in 1959. This idea reduces the category of irreducible actual representations of a true Lie algebra to an outline of the so-called self-conjugate irreducible advanced representations of this algebra and to the calculation of an invariant of this sort of illustration (with values $+1$ or $-1$) referred to as the index.

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14 e. I varieties because, for a n y b a s i c homomoprhism starting For from e : Z ~ K, Z, m a y be instance, the e l e m e n t a r y and m o r p h i s m s said S y m n ( X l,x 2 ..... 19 T. e. c. m. o. r. p. h. i. s m morphism, to be deserve there a special is a u n i q u e the c h a n g e s o f rings, automatic. Symn: = Dn ~ Dn is d e f i n e d (Xix i , Z i < j x i x j ..... x l - , . t theorem. e. a morphism f(x I ..... 18 symmetric ring so that the m o r p h i s m symmetric over f : Dn ~ V verifying (I) .....

T theorem. e. a morphism f(x I ..... 18 symmetric ring so that the m o r p h i s m symmetric over f : Dn ~ V verifying (I) ..... 20 of the g reduces a formal module out I ..... n such . 17) , the b a s i c V, this symmetric ring K may theorem polynomials. remain unde- fined. CHAPTER FORMAL GROUPS I. 2 admits ables on G corresponding tities G ways in the c a t e g o r y , alternative by giving, theory qroups, products. with some of group (resp. (resp. commutative G n ~ G, word-functions or c o m m u Anyhow, extra a struc- descriptions.

Now, ~(1) ~)K1 • rinqs series , fj(x) unchanged, presently we denote the same f : D K(I) ~ D(J) K have We there ~ DKt of m o d e l s over K de fin in q precisely, f. M o r e = E~ c 3 , ~ x (~o~). V) (A') e*A' ~ nil(K) variety for a n y )X to e v e r v , j ~ J, functor the by 9. o~. the c a t e g o r y over K'. Note that , ~,d~ . functor variety over A'~ the K - a l g e b r a A' , m u l t i p l i c a t i o n corresponds e. to the c a t e g o r y K. 2 vectors . (1) of the. f o r m a l (fj)j~j will K' w i t h To e v e r y m o r p h i s m by applyinq taDgent ~'V .