Download Abstract Lie Algebras by David J Winter PDF

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. wintry weather, a Professor of arithmetic on the collage of Michigan, additionally offers a common, broad therapy of Cartan and similar 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 contain solvable and nilpotent Lie algebras, Cartan subalgebras, and Levi's radical splitting theorem and the whole 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 concept of Cartan and similar subalgebras of Lie algebras over arbitrary fields is constructed within the final...

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Extra resources for Abstract Lie Algebras

Example text

The first assertion is obvious, as is one direction of the second. Next, let be –completely reducible. Then where the are -irreducible -submodules of . Thus, the are ideals of . We claim that they are simple. Thus, note that for i ≠ j, since . Thus, an ideal of is an ideal of , and . It follows that has no proper ideal and for all i. 5 Corollary Let be finite dimensional and where the are simple ideals of . Then every ideal of is a sum of certain of the . In particular, the are the only simple ideals of .

As in the theory of groups, is solvable iff there exists a series of ideals such that is Abelian for all i; similarly, is nilpotent iff there exists a series such that is central in for all i. 3 Definition A representation of a Lie algebra is a homomorphism f from into (Homk V)Lie for some finite-dimensional vector space V over k. 4 Definition A Lie module for is an -module over k such that m(xy) = (mx)y — (my)x for If is a representation of , then V together with the module operation mx = mf(x) for is a Lie module for .

2 Proposition is associative (respectively Lie) iff is associative (respectively Lie). PROOF. 3. 1. 3 Proposition Let be subspaces of . Then . ) PROOF. Let {ei}, {fj} be bases for over k. Then {ei}, {fj} are bases for over k′. Now {eifj} is a generating set for over k′as well as for over k′. Thus, . 4 Definition Let be a nonassociative algebra over k′, k a subfield of k′ and a nonassociative algebra over k. Then is a k-form of if the underlying vector space of is a k-form of the underlying vector space of and, for x, , m(x, y) = m′(x, y), where m, m′ are the product mappings for respectively.

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