ON A PARABOLIC ALGEBRA-P OF U-Q(SL(N+L)) AND ITS SEMIVARIANTS BY THE ADJOINT ACTION OF P

In this article, we prove the quantum analogue of some results shown i n the classical case by J. DIXMIER ([Dix]) and A. JOSEPH ([J2]). We co nstruct two subalgebras G and P of U-q(sl(n+1)) such that U-q(sl(n+1)) similar or equal to G x P as vector spaces and such that the generato rs of the centre of U-q(sl(n+1)) are expressed as a linear combination of generators of the algebra G with coefficients in P. Moreover, cons idering the vector space generated by the semi invariants of P with re spect to its adjoint action, we prove that it is a polynomial algebra of one variable ct and we give a description of d similar to the descr iption given by A. JOSEPH ([J2], 7.4.) (C) Elsevier, Paris.