# Algebra Carbondale 1980 proceedings. Lie algebras, group by Ralph K. Amayo

By Ralph K. Amayo

Those are the court cases of the Southern Illinois Algebra Conference.The convention was once held on the Southern Illinois college at Carbondale on April 18 and 19, 1980 as a part of the specific Week In Algebra , April14-19, 1980. The convention was once made attainable during the curiosity, monetary help and cooperation of the dept of arithmetic, university of Liberal Arts, Graduate college and workplace of foreign schooling at Southern Illinois college at Carbondale.

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Phys . , Reshetikhin N. Yu . : 1990 Leningrad Math . J. , Sorace E . : 1992 Contractions o] quan tum groups (Lecture Notes in Mathemati cs 1510) (Berlin : Springer) p . , Ruegg H. : 1992 Phys . L ett . B . , Sorace E . : 1992 Phys. Rev. Lett. 68 3178 Inonii E . : 1953 Pro c. N atl. d. Sci . U. S. J. : 1993 Proc. XIX ICGTMP CIEMAT/RSEF (Madrid) Vol. I, p. 455. : 1990 Int. J. Mod . Phys . A 5 1 QUANTUM UNIVERSAL ENVELOPING CAYLEY-KLEIN ALGEBRAS 23 13. Truini P. : 1993 Re v. Math. Phys . 5 363 14.

We will prove where aj is equal to e, Uj E U or z; E Z . 44 J . B ERNAD ET AL. Let us suppose aj E N = ker w Vi. ''''e can assu me aj E N = N / ann(N ), so (7) By (7), RuRal ·· · RakRu = - n; «; · ·· Rak- l RuRak - RuRal · · · Rak- l Ruak Repeting (7) several ti mes, we obtain where between Cl , ' • . aj E N2 . Since R; = -~Ru2,we have poved that RuRal" . RakRu is a sum of products Rbi with at least one element bj E N 2 in each sumand. Let us see that if T = RuoRuo ··· R; is a sum of products Rbi and in each sumand there are k elemen ts UG j E N 2 , adding to T operators DRu we obtain T' = 7DRu .

And consider T R; . . R a .. R u where T has the same right operators except R e . It is easy to show that if T 0, then T = O. This means that we can remove operators R, to prove the nilpotency 0 of idl(Ru) . Then, by the first case, idl(R u ) is nilpot ent . 2 Let (B,w) be a nuclear Bernstein algebra , characteristic :f; 2, 3, 5, and Z E U 2 . The ideal idl( R z )