On some applications of the universal enveloping algebra of a semisimple Lie algebra

Now we come to the second question.Let us call an element X£g rational if \(Hi)ÇzKo, l^iél, where K0 is the prime field of characteristic zero contained in K.It is clear that X = 2^iá«g¡ c0 ifXf^O and c¿>0, i being the least index such that Cj^O.Let go be the set of all rational linear functions on f).For any X, p-Ggo we write X>p, or p. 0.In this way go is completely ordered under the relation >.This order is called the lexicographic order in go with respect to the ordered set {«i, • • • , at}.We shall call an element X£g0 integral if X(i/¿) is an integer HARISH-CHANDRA [Januarywhere StJ-is the usual Kronecker symbol.Since 21' is a free algebra and Vh ' ' " Vir< 1 áji.• ■ • , jrúh r^O, form a base for ® it is clear that 7r"' is uniquely defined by the above equations.It is easily verified that the kernel of irj! contains the set© and therefore IF.Now suppose H' G £)' and ir¿ (IT) = 0.