# Nilpotent Groups as Perturbations of the Identity

In Terry Tao's blog, specifically this post, he says that nilpotent groups can be thought of in algebraic geometry and analysis as modeling infinitesimal perturbations of the identity. Can someone provide a reference where this point of view is used?

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If you can read french, have a look at this. numdam.org/numdam-bin/browse?id=PMIHES_1965__26_ –  Baby Dragon Jul 3 '12 at 7:36

The idea of nilpotents representing infinitesimal change dates back at least to the invention/discovery of calculus by Newton and Leibniz! In deformation theory of schemes at least, the whole story begins with the study of flat families over the spectrum of $k[t]/t^2,$ which are viewed as infinitesimal perturbations of the "central" fibre parametrized by the nilpotent parameter $t.$ This is mentioned in Hartshorne II.8 in the exercises I believe, but there is an entire field of work in deformation theory dedicated to this beautiful concept. For example, Harthorne has a recent introductory book called "Deformation Theory".