{"product_id":"lie-algebras-and-lie-groups-jean-pierre-serre-9783540550082","title":"Lie Algebras and Lie Groups: 1964 Lectures Given at Harvard University","description":"The main general theorems on Lie Algebras are covered, roughly the content of Bourbaki's Chapter I. I have added some results on free Lie algebras, which are useful, both for Lie's theory itself (Campbell-Hausdorff formula) and for applications to pro-Jrgroups. of time prevented me from including the more precise theory of Lack semisimple Lie algebras (roots, weights, etc.); but, at least, I have given, as a last Chapter, the typical case ofal, .. This part has been written with the help of F.Raggi and J.Tate. I want to thank them, and also Sue Golan, who did the typing for both parts. Jean-Pierre Serre Harvard, Fall 1964 Chapter I. Lie Algebras: Definition and Examples Let Ie be a commutativering with unit element, and let A be a k-module, then A is said to be a Ie-algebra if there is given a k-bilinear map A x A A (i.e., a k-homomorphism A0\" A -] A). As usual we may define left, right and two-sided ideals and therefore quo- tients. Definition 1. A Lie algebra over Ie isan algebrawith the following properties: 1). The map A0i A -+ A admits a factorization A (R)i A -+ A2A -+ A i.e., ifwe denote the imageof(x, y) under this map by  x, y) then the condition becomes for all x e k.  x, x)=0 2). (lx, II], z]+ny, z), x) + ( z, xl, til = 0 (Jacobi's identity) The condition 1) implies  x,1\/]=- 1\/, x).\u003cbr\u003e\u003cbr\u003e\u003cb\u003eAuthor:\u003c\/b\u003e Jean-Pierre Serre\u003cbr\u003e\u003cb\u003eISBN-10:\u003c\/b\u003e 3540550089\u003cbr\u003e\u003cb\u003eISBN-13:\u003c\/b\u003e 9783540550082\u003cbr\u003e\u003cb\u003ePublisher:\u003c\/b\u003e Springer\u003cbr\u003e\u003cb\u003eLanguage:\u003c\/b\u003e English\u003cbr\u003e\u003cb\u003ePublished:\u003c\/b\u003e 03\/11\/1992\u003cbr\u003e\u003cb\u003ePages:\u003c\/b\u003e 173\u003cbr\u003e\u003cb\u003eFormat:\u003c\/b\u003e Paperback\u003cbr\u003e\u003cb\u003eWeight:\u003c\/b\u003e 0.60lbs\u003cbr\u003e\u003cb\u003eSize:\u003c\/b\u003e 9.10h x 6.10w x 0.40d","brand":"Jean-Pierre Serre","offers":[{"title":"Paperback","offer_id":44125993074943,"sku":"9783540550082","price":44.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0662\/2982\/9887\/files\/img_8a610255-095c-4c9a-bcc7-9b37d00c5847.jpg?v=1687445694","url":"https:\/\/www.whiterainbookhouse.com\/products\/lie-algebras-and-lie-groups-jean-pierre-serre-9783540550082","provider":"WR Book House","version":"1.0","type":"link"}