By David J Winter
Solid yet concise, this account of Lie algebra emphasizes the theory's simplicity and provides new ways to significant theorems. writer David J. wintry weather, a Professor of arithmetic on the collage of Michigan, additionally provides a normal, wide remedy of Cartan and similar Lie subalgebras over arbitrary fields.
Preliminary fabric covers modules and nonassociate algebras, through a compact, self-contained improvement of the speculation of Lie algebras of attribute zero. themes comprise solvable and nilpotent Lie algebras, Cartan subalgebras, and Levi's radical splitting theorem and the entire reducibility of representations of semisimple Lie algebras. extra matters 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 in depth idea of Cartan and comparable subalgebras of Lie algebras over arbitrary fields is built within the final...
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Additional resources for Abstract Lie Algebras
1 Definition An -module over k is a vector space over k together with a mapping , denoted , such that (αm + βn)s = α(ms) + β(ns) for , and . 2 Definition Let be an -module over k, . Then TM is the linear transformation of defined by mTM = mT for . 3 Definition The direct sum of -modules over k is the -module with underlying vector space the direct sum of the vector spaces together with the mapping defined by For an -module over k and , we let be the subspace of generated by the set . 4 Definition An -submodule of an -module is a subspace of such that .
The following discussion is relative to these fixed k-forms. 9 Definition An element is defined over k if . 10 Proposition Let be finite dimensional over k′(i = 1, 2) and let be defined over k. Then Kernel T and Image T are defined over k. PROOF. is defined over k, as the k′-span of . Now . Thus, Kernel spans Kernel T over k′ and Kernel T is defined over k. 11 Definition is defined over k}. 12 Proposition Let be finite dimensional over k′ (i =1, 2). Then is a k-form of and the mapping given by is a k-isomorphism.
8 Corollary Let be finite dimensional. Then has a unique maximal solvable ideal. PROOF. Let , be maximal solvable ideals of . 7, so that . 9 Definition The maximal solvable ideal of a finite-dimensional nonassociative algebra is called the radical of and is denoted Rad . If is a finite-dimensional associative algebra, the radical of is the maximal nilpotent ideal of . Even in this case, however, Rad need not be a characteristic ideal of . 10 Example Let be an algebra over a field k of characteristic p > 0 with basis 1, x, …, xp − 1 such that xp = 1.
Abstract Lie Algebras by David J Winter