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.

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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,

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