Algebraic aspects of the theory of multiplications in complex cobordism theory

Abstract

The general classification problem for stable associative multiplications in complex cobordism theory is considered. It is shown that this problem reduces to the theory of a Hopf algebra S (the Landweber-Novikov algebra) acting on the dual Hopf algebra S* with distinguished ”topologically integral” part Λ that corresponds to the complex cobordism algebra of a point. We describe the formal group and its logarithm in terms of the algebra representations of S. The notion of one-dimensional representations of a Hopf algebra is introduced, and examples of such representations motivated by well-known topological and algebraic results are given. Divided-difference operators on an integral domain are introduced and studied, and important examples of such operators arising from analysis, representation theory, and non-commutative algebra are described. We pay special attention to operators of division by a non-invertible element of a ring. Constructions of new associative multiplications (not necessarily commutative) are given by using divided-difference operators. As an application, we describe classes of new associative products in complex cobordism theory.