# Lie Algebras and Lie Groups: 1964 Lectures given at Harvard by Jean-Pierre Serre

By Jean-Pierre Serre

This publication reproduces J-P. Serre's 1964 Harvard lectures. the purpose is to introduce the reader to the "Lie dictionary": Lie algebras and Lie teams. specified positive aspects of the presentation are its emphasis on formal teams (in the Lie staff half) and using analytic manifolds on p-adic fields. a few wisdom of algebra and calculus is needed of the reader, however the textual content is well available to graduate scholars, and to mathematicians at huge.

Similar linear books

A first course in linear algebra

A primary path in Linear Algebra is an advent to the elemental suggestions of linear algebra, besides an creation to the thoughts of formal arithmetic. It starts off with structures of equations and matrix algebra sooner than getting into the idea of summary vector areas, eigenvalues, linear alterations and matrix representations.

Measure theory/ 3, Measure algebras

Fremlin D. H. degree idea, vol. three (2002)(ISBN 0953812936)(672s)-o

Elliptic Partial Differential Equations

Elliptic partial differential equations is among the major and so much energetic parts in arithmetic. In our publication we research linear and nonlinear elliptic difficulties in divergence shape, with the purpose of supplying classical effects, in addition to more moderen advancements approximately distributional suggestions. as a result the ebook is addressed to master's scholars, PhD scholars and somebody who desires to start study during this mathematical box.

Additional resources for Lie Algebras and Lie Groups: 1964 Lectures given at Harvard University

Example text

3. If 9 U ,olvable and k of characteri,tic zero, then (g, gl i, nilpotent. Since the statement is linear, we may suppose that k is algebraically closed. (If k' is an extension field of k, and g' = lJ ®k k', then it is obvious that 9 is solvable (resp. nilpotent) if and only if S' is solvable (resp. ) By the preceding corollary there is a flag (gi) of ideals in 9, say 9 ::> 91 :::> 92 :::> ••• :::> gn = O. Let x E (g, g]. Then ad z 9i C gi+l because End(li/gi+l) ~ k is commutative. Hence adz is nilpotent on I, and all the more so on (1,9].

H2 , so we have h 1 h 2 Second ClUe. h 2 ~ X. , ha,h. ]] = hl (hah4 ) b) hi <;: h a E H and < h•. • < h•. , [hI', ha)] · Now length of h 1 h. ) = Lealia where hOI E H. )which implies t(hOl ) > f(hl ), hence hOI > hi. &a, hOI) > hI = lnf(h1 , h2 ). ' Applying the induction hypothesis we see that (ha , hal is a linear combination of b's with h E H. ,[h1 ,ha]) is also a linear combination of h's with h e H. d. = 6. ) Let X be a set and let Fx be the free group on X. Let Fl be the descending central series of Fx, defined by F} = Fx and F (Fx,F;-l), for n > 1.

Let cPa be the restriction of tP to Ga. Claim: There is an index {J such that 4>fJ : Gil ~ a is an isomorphism, and tPa = 0 for a :/= {J. For each a, the image of 4>aClOl is an ideal in _, because cP is surjective and CIa an ideal in 9. Hence by the simplicity of a,