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Let Z; (resp. Z;) be the subalgebra generated by the yt (resp. Toyt), k = I , ... , N. These are polynomial algebras (independent of the choice of J; see §3.3) and we have -0 II 12 IN_I J Lemma. s. S. ···S. (a-,) = a .. II 12 IN_I J J Proof. s. s· .. ·s· (a-,) = s.wo(a-,) = s.(-a.). 0 II 12 IN_I J J \ J J J -II 12 Ik_1 J J Yk := T. T. ... T. (y.) = Tk_l(y· ). J J' Let k -1 = l(a). In case (1) we remark that Yh = Yh if h =Ik, k + 1, J J ' . . J J' Yk = Ta(y j) = TaTj(Y j) = Yk+1 ' and simllarlydoi:10.1090/s0894-0347-1992-1124981-x fatcat:7rxvjofj5bhmldwbfptlkc4n4m