Sophus Lie's 1884 Differential Invariant Paper

Description

S. Lie "Uber Differentialinvarianten" [Math. Ann. 24 (1884), 537–578]

This is the second volume in the above series to focus on the work of Sophus Lie. Volume I was devoted to a translation and exposition of a paper of 1880 by Lie in which the fundamental concepts of his theory were introduced and applied to the problem of determining all groups in one or two variables. Applications of his theory to differential equations were always foremost in his mind, and the present volume contains a reliable translation of one of his many papers dealing with differential equations. It is concerned primarily with the differential invariants of infinite-dimensional groups.
Intertwined with the translation is a running commentary by Robert Hermann in which he "translates'' the text into the idiom of contemporary mathematics and examines some of its implications. In addition to the commentary, he has written an introductory chapter ("Contact manifolds and mapping element spaces'') summarizing the contemporary notions he employs in the commentary, and several concluding chapters, which "contain new or improved material that can serve as a basis for further research''. These chapters include Chapter D ("Differential Invariants and Groups of Symmetries of Differential Equations''), which "contains a general approach to Lie's whole work on differential equations and symmetries'', and Chapter E, which develops related ideas on the "Galois theory'' of differential equations and includes a sketch of such a theory beyond the well-known Picard-Vessiot case. Hermann's motivation for presenting Lie's work is neither antiquarian nor even, primarily, historical. He had his eye upon the future when he wrote in the preface: "In reading Lie's work in preparation for my commentary on these translations, I was overwhelmed by the richness and beauty of the geometric ideas flowing from Lie's work. Only a small part of this has been absorbed into mainstream mathematics. He thought and wrote in grandiose terms, in a style that has now gone out of fashion, and that would be censored by our scientific journals! The papers translated here and in the succeeding volumes of our translations present Lie in his wildest and greatest form.'' And therein lies the challenge to present-day mathematicians.

Reviewed by Thomas Hawkins