{"product_id":"lie-algebras-and-lie-groups-book-jean-pierre-serre-9783540550082","title":"Lie Algebras and Lie Groups","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).","brand":"WoB","offers":[{"title":"GB \/ NEW \/ GARDNERS","offer_id":50683299922193,"sku":"NGR9783540550082","price":47.89,"currency_code":"USD","in_stock":false},{"title":"US \/ NEW \/ INGRAM","offer_id":51064444322065,"sku":"NIN9783540550082","price":44.99,"currency_code":"USD","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":52544278495505,"sku":"NLS9783540550082","price":52.89,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/3540550089.jpg?v=1751253030","url":"https:\/\/www.worldofbooks.com\/products\/lie-algebras-and-lie-groups-book-jean-pierre-serre-9783540550082","provider":"World of Books ","version":"1.0","type":"link"}