Algebraic aspects of the theory of multiplications in complex cobordism theory

The general classification problem for stable associative multiplications i n complex cobordism theory is considered. It is shown that this problem red uces to the theory of a Hopf algebra S (the Landweber-Novikov algebra) acti ng on the dual Hopf algebra S* with distinguished 'topologically integral' part Lambda that corresponds to the complex cobordism algebra of a point. W e describe the formal group and its logarithm in terms of the algebra repre sentations of S. The notion of one-dimensional representations of a Hopf al gebra is introduced, and examples of such representations motivated by well -known topological and algebraic results are given. Divided-difference oper ators on an integral domain are introduced and studied, and important examp les of such operators arising from analysis, representation theory, and non -commutative algebra are described. We pay special attention to operators o f division by a non-invertible element of a ring. Constructions of new asso ciative multiplications (not necessarily commutative) are given by using di vided-difference operators. As an application, we describe classes of new a ssociative products in complex cobordism theory.