{"product_id":"lie-algebras-and-lie-groups-1964-lectures-given-at-harvard-university-9783540550082","title":"Lie Algebras and Lie Groups: 1964 Lectures Given at Harvard University","description":"\u003cp\u003e • Author(s): Jean-Pierre Serre\u003cbr\u003e • Publisher: Springer\u003cbr\u003e • Publisher Imprint: Springer\u003cbr\u003e • BISAC: Algebra - Linear\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e\u003cp\u003eThe 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).\u003c\/p\u003e","brand":"Springer","offers":[{"title":"Paperback","offer_id":45035016683671,"sku":"9783540550082","price":2938.0,"currency_code":"INR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0666\/3471\/1191\/files\/9783540550082.webp?v=1767262430","url":"https:\/\/atlanticbooks.com\/products\/lie-algebras-and-lie-groups-1964-lectures-given-at-harvard-university-9783540550082","provider":"Atlantic Books","version":"1.0","type":"link"}