Algebraic Aspects of the Theory of Product Structures in Complex Cobordism

We address the general classification problem of all stable associative product structures in the complex cobordism theory. We show how to reduce this problem to the algebraic one in terms of the Hopf algebra S (the Landweber-Novikov algebra) acting on its dual Hopf algebra S∗ with a distinguished “topologically integral” part Λ that coincids with the coefficient ring of the complex cobordism. We describe the formal group and its logarithm in terms of representations of S. We introduce one-dimensional representations of a Hopf algebra. We give series of examples of such representations motivated by well-known topological and algebraic results. We define and study the divided difference operators on an integral domain. We discuss certain important examples of such operators arising from analysis, representation theory, and noncommutative algebra. We give a special attention to the division operators by a noninvertible element of a ring. We give new general constructions of associative product structures (not necessarily commutative) using the divided difference operators. As application, we describe new classes of associative products in the complex cobordism theory.