Titlebook: Introduction to Lie Algebras and Representation Theory; James E. Humphreys Textbook 1972 Springer-Verlag New York Inc. 1972 Lie.algebra.al

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James E. HumphreysIncludes supplementary material:

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Representation Theory,root system, Δ= {α.,…,α.} a base of Φ,. if the Weyl group. The main object is to study finite dimensional .-modules (although certain infinite dimensional modules will also appear). Thanks to Weyl’s Theorem on complete reducibility, it is the irreducible modules which play a controlling role in the finite dimensional case.

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Root Systems,y to write down an explicit formula: . (This works, because it sends α to — α and fixes all points in ..) Since the number 2(., α)/(α, α) occurs frequently, we abbreviate it by <., α>. Notice that <., α> is linear only in the first variable.

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Isomorphism and Conjugacy Theorems, where l = dim... By extending the base field from . to . we therefore obtain an l-dimensional real vector space E spanned by Φ. Moreover, the symmetric bilinear form dual to the Killing form is carried along to E, making E a euclidean space. Then Theorem 8.5 affirms that Φ is a root system in E.

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Existence Theorem,ugh it could have been introduced right away in Chapter I, we deferred it until now in order to avoid the unpleasant task of proving the Poincaré-Birkhoff-Witt Theorem before it was really needed. The reader is advised to forget temporarily all the specialized theory of semisimple Lie algebras.

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