Abelian and nonabelian mathematics

Abstract

Many theories that are part of the most beautiful chapters in the mathematics of the nineteenth and twentieth centuries have a common algebraic nucleus: the duality theory of abelian groups and their groups of characters. It is natural to assume that a series of basic difficulties in mathematics can be explained by the lack of a generalization of this duality to nonabelian groups. Such a point of view has been presented in A. Weil's paper [6]. The "nonabelian mathematics of the future" philosophy also inspired me when I was just starting to work on mathematics. Now, half a century later, I think it would be interesting to sum up the achievements in this direction, the more so since, just as before, the unsolved problems ou tnumber the solved ones, so that the old program may apparently inspire yet another generation. The goal of the following survey is to give a general idea of the intertwining of ideas that arise in this area. The most important example of a pairing is provided by the integration of differential forms. If, for example, 00 is a 1-form, and r is a curve on a manifold X, then f~00 can be considered as a pairing (r More precisely, the pairing is defined between the space H~(X;R) of one-dimensional homologies and the quotient space {o~ : d00 = 0}/{00 = df}. The nondegeneracy of this pairing and its generalization to m-dimensional chains is guaranteed by the de Rham theorem. (One could also substitute C for R everywhere.) A nondegenerate symmetric form on V determines an isomorphism V* -~ V. If the real vector space V is oriented, then the operation W ~ W • carries over to