Lie Algebras: Theory and Algorithms
The aim of the present work is two-fold. Firstly it aims at a giving an account of many existing algorithms for calculating with finite-dimensional Lie algebras. Secondly, the book provides an introduction into the theory of finite-dimensional Lie algebras. These two subject areas are intimately related. First of all, the algorithmic perspective often invites a different approach to the theoretical material than the one taken in various other monographs (e.g., , , , ). Indeed, on various occasions the knowledge of certain algorithms allows us to obtain a straightforward proof of theoretical results (we mention the proof of the Poincaré-Birkhoff-Witt theorem and the proof of Iwasawa's theorem as examples). Also proofs that contain algorithmic constructions are explicitly formulated as algorithms (an example is the isomorphism theorem for semisimple Lie algebras that constructs an isomorphism in case it exists). Secondly, the algorithms can be used to arrive at a better understanding of the theory. Performing the algorithms in concrete examples, calculating with the concepts involved, really brings the theory of life.
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Chapter 3 Cartan subalgebras
Chapter 4 Lie algebras with nondegenerate Killing form
Chapter 5 The classification of the simple Lie algebras
Chapter 6 Universal enveloping algebras
Chapter 7 Finitely presented Lie algebras
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algebra of characteristic algorithm for calculating associative algebra automorphism called Cartan matrix Cartan subalgebra coefficient computing construct contains Corollary corresponding defined deg(m denote direct sum dominant weight Dynkin diagram eigenvalues Example field F field of characteristic finite-dimensional Lie algebra follows Furthermore Gröbner basis ground field Hall elements Hall set Hall word Hence highest weight idempotents implies induction input integer isomorphic Jacobi identity Killing form L-module left factor Lemma Let f Let g Let H Levi subalgebra linear combination linear map LM(f LM(g minimum polynomial modulo G monomials multiplication table nilpotent ideal non-zero NR(L output positive roots primary decomposition Proof Proposition prove Rad(A relative Rlex Root fact root space root system Section semisimple Lie algebra simple Lie algebra simple roots simple system solvable radical spanned splitting element SR(L standard monomials Step straightforward structure constants submodule subspace suppose Theorem vector space