By David J Winter

Solid yet concise, this account of Lie algebra emphasizes the theory's simplicity and gives new methods to significant theorems. writer David J. wintry weather, a Professor of arithmetic on the collage of Michigan, additionally offers a basic, vast remedy of Cartan and comparable Lie subalgebras over arbitrary fields.

Preliminary fabric covers modules and nonassociate algebras, by way of a compact, self-contained improvement of the speculation of Lie algebras of attribute zero. subject matters comprise solvable and nilpotent Lie algebras, Cartan subalgebras, and Levi's radical splitting theorem and the full reducibility of representations of semisimple Lie algebras. extra matters comprise 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 idea of Cartan and comparable subalgebras of Lie algebras over arbitrary fields is constructed within the final...

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Since the sums and are direct, it follows that is direct. 12 Definition If is called the weight space for in with respect to a and a is called a weight of in Since is finite dimensional, has only finitely many weights in . 13 Theorem Let be a nilpotent subalgebra of and let be a finite-dimensional -module. Regard as an -module via ad. Then for all functions . PROOF. Consider the split extension of by , having the product . Then and where a′ = a ad. 12, and since is an ideal of for all a′, b′. 14 Corollary Let be a nilpotent ideal in , and let be a finite-dimensional -module.

Since is a nilpotent ideal of is a nonzero -submodule of and by the -irreducibility of . ) Thus, . Since is central. 5. Thus, . 7 Theorem (Lie) Let be a solvable Lie algebra over k, and let be a finite-dimensional irreducible Lie module for over k. Suppose that has a Cartan sub-algebra such that every is split on over k. Then dim . PROOF. Assume that ≠ {0}. 3. Now is an -submodule, so , by the -irreducibility of . Thus, and we may regard as an -module, . Now is Abelian and , so and . Thus, . Since every element of is split over k, every element of is split over k.

Next, assume that for α, β ∈ k. Let . Then and we have (xy)D = (α + β)xy = (αx)y + x(βy) = (xD)y + x(yD). Since spans , it follows that . 14 Corollary Let , where is finite dimensional. Then if D is split or separable over k, Ds and Dn are contained in Der . PROOF. Suppose first that D is split. 13. Thus, . Next, suppose that D is separable over k, let k′ be an algebraic Galois extension of k such that D is split over k′ and let . Then , so that Ds, by the above paragraph. 10. Thus, Ds, . 15 Proposition Let D be a nilpotent derivation of , where is finite dimensional over k and k is of characteristic 0.