## Abstract Lie AlgebrasSolid but concise, this account of Lie algebra emphasizes the theory's simplicity and offers new approaches to major theorems. Author David J. Winter, a Professor of Mathematics at the University of Michigan, also presents a general, extensive treatment of Cartan and related Lie subalgebras over arbitrary fields. Preliminary material covers modules and nonassociate algebras, followed by a compact, self-contained development of the theory of Lie algebras of characteristic 0. Topics include solvable and nilpotent Lie algebras, Cartan subalgebras, and Levi's radical splitting theorem and the complete reducibility of representations of semisimple Lie algebras. Additional subjects include 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 extensive theory of Cartan and related subalgebras of Lie algebras over arbitrary fields is developed in the final chapter, and an appendix offers background on the Zariski topology. |

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6-module 6-submodule A-graded Lie algebra algebra respectively Lie algebraically closed algebras of characteristic associative algebra automorphism Cartan subalgebra Choose classiﬁcation closed under pth commute consists of nilpotent contains Corollary Let decomposition deﬁned Deﬁnition Deﬁnition Let denoted dimensional direct sum Dynkin diagram Engel subalgebra exists extension ﬁeld ﬁnite ﬁnite-dimensional Lie ﬁnite-dimensional vector space follows half-system homomorphism ideal of 91 induction on dim inﬁnite irreducible isomorphism k-form k’-span Let f Lie algebra respectively Lie module Lie p-module linear transformation mapping maximal solvable subalgebra maximal torus minimal nilpotent subalgebra nonassociative algebra nonzero PROOF Proposition Let pth powers quasi-regular reﬂection semisimple Lie algebras simple Lie algebra simple system 11 split Cartan subalgebra split semisimple Lie stable standard subset subalgebra of 53 subgroup subspace Theorem Let vector space weak root system Weyl group Zariski topology