I have been interested in analytic groups and algebraic structures on them for some time – ever since an anonymous reviewer of a grant proposal suggested I take a look at some papers of Hochschild and Mostow in connection with the work I proposed. I found the papers fascinating, and useful as I began work in the area, but the foundations of the subject seemed to me (as an autodidact Lie theorist) irrevocably recondite. Then, in the summer of 1978, two things happened: while preparing for a talk for a conference in Copenhagen, I found elementary arguments for the basic existential facts of the theory (these appear here in Chapter 3) and, more important, I met Alex Lubotzby at the conference, who pointed out the connection between the Hochschild–Mostow theory and the Grothendieck theory of Tannakian categories (this appears here in Chapter 2). These things meant that the subject was both easier and of wider application than I had previously imagined, and so inspired this volume.

The book was written while I enjoyed a sabbatical leave from The University of Oklahoma and was Visiting Professor of Mathematics, University of Virginia (Fall 1979), Visiting Professor of Mathematics, Bar-Ilan University (Winter 1980), and Visiting Professor of Mathematics, University of California, Berkeley (Spring 1980). To those institutions and my colleagues there, I am grateful. I especially want to acknowledge the encouragement I have gotten over the years from Gerhard Hochschild.