Abstract Lie Algebras by David J Winter

By David J Winter

Solid yet concise, this account of Lie algebra emphasizes the theory's simplicity and gives new ways to significant theorems. writer David J. iciness, a Professor of arithmetic on the collage of Michigan, additionally provides a common, large remedy of Cartan and similar Lie subalgebras over arbitrary fields.

Preliminary fabric covers modules and nonassociate algebras, via a compact, self-contained improvement of the idea of Lie algebras of attribute zero. themes contain solvable and nilpotent Lie algebras, Cartan subalgebras, and Levi's radical splitting theorem and the total reducibility of representations of semisimple Lie algebras. extra topics contain the isomorphism theorem for semisimple Lie algebras and their irreducible modules, automorphism of Lie algebras, and the conjugacy of Cartan subalgebras and Borel subalgebras. an intensive concept of Cartan and similar subalgebras of Lie algebras over arbitrary fields is constructed within the final...

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Thus, we may assume that is a subalgebra of (Homk )Lie and f(x) = x for . Let be a Cartan subalgebra of , where the a’s are roots of and . Let , , h = [x, y]. Then 0 = (h, h) = Trace h2 since . 1. Thus, is nilpotent} for all a. 4. Thus, is an ideal of . If then is solvable, by induction, and is solvable, so that is solvable. Otherwise . Then . 8. Then and is again solvable. The above proof would be more natural if we knew that the ideal is nilpotent, for we would not need an auxiliary argument for the case .

2 Definition The ideals , i ≥ 1, are recursively defined by for i ≥ 2. 3 Definition is solvable if for some i. 4 Definition is nilpotent if for some i. If is associative, is solvable if and only if is nilpotent, and the notion of nilpotency is equivalent to the usual notion of nilpotency. 5 Proposition Let k′ be an extension field of k. Then is solvable (respectively nilpotent) iff is solvable (respectively nilpotent). PROOF. 3. Thus, , i ≥ 0, and is solvable iff is solvable. Similarly, , i ≥ 1, and is nilpotent iff is nilpotent.

Since , there is a refinement which is x-stable and -stable, with for all i. Thus, . By the maximality of , and Thus, . 5 Corollary Let be a subalgebra of (Homk )Lie, where is finite dimensional over k. If is a solvable ideal of , then the set of nilpotent elements of is an ideal of , and . If is a nilpotent ideal of , then consists of nilpotent elements. PROOF. Suppose that is a solvable ideal of . Let x ∈ and . 4. Since this is true for any . 4, it follows that is an ideal of . Now let be a nilpotent ideal of and let .

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