{ "item_title" : "Lie Algebras and Lie Groups", "item_author" : [" Jean-Pierre Serre "], "item_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 byx, 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) impliesx,1/]=- 1/, x).", "item_img_path" : "https://covers3.booksamillion.com/covers/bam/3/54/055/008/3540550089_b.jpg", "price_data" : { "retail_price" : "44.99", "online_price" : "44.99", "our_price" : "44.99", "club_price" : "44.99", "savings_pct" : "0", "savings_amt" : "0.00", "club_savings_pct" : "0", "club_savings_amt" : "0.00", "discount_pct" : "10", "store_price" : "" } }

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