An Algebra of Skew Primitive Elements

We study the various term operations on the set of skew primitive elements of Hopf algebras, generated by skew primitive semi-invariants of an Abelian group of grouplike elements. All 1-linear binary operations are described and trilinear and quadrilinear operations are given a detailed treatment. Necessary and sufficient conditions for the existence of multilinear operations are specified in terms of the property of particular noncommutative polynomials being linearly dependent and of one arithmetic condition. We dub the conjecture that this condition implies, in fact, the linear dependence of the polynomials in question and so is itself sufficient ( a proof of this conjecture see in ”An existence condition for multilinear quantum operations,” Journal of Algebra, 217, 1999, 188− 228).