Graded Dual of the Non-Commutative Universal Leibniz Hopf Algebra Zeta

摘要

In the paper the authors consider the dual Z and the graded dual M of theuniversal Leibniz Hopf algebra Z = Z < Z(sub 1), Z(sub 2),... > (i.e., the free associative (but non-commutative) algebra over Z on the set (Z(sub 1), Z(sub 2),...). The authors shall show that M = Z(EUW) as algebras, where Z(EUW) is the free polynomial ring over Z generated by the set EUW. P.B. Shay showed in his preprint that, without giving generic variables, Z is a formal power series ring over Z. Hence, it easily follows that M is a polynomial ring. The difficult part of proving the authors' statement is that they have to determine the set of (algebraically independent) generic variables, i.e. the set EUW. However, the corollaries to their main result are worth the effort. For instance, it immediately follows that Z is a non-commutative formal group law over Z. Another corollary is that they can give a Z-module basis for the primitive elements of Z.