Lie Algebras: Theory and Algorithms

by de Graaf

• ISBN13: 9780444501165
• ISBN10: 0444501169
• eBook ISBN(s): 9780080535456, 9780444501165
• Format: Hardcover
• Copyright: 2000-02-04
• Publisher: Elsevier Science
• More Book Details
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Summary

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., [42], [48], [77], [86]). 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.

Table of Contents

Basic constructions	1	(39)
Algebras: associative and Lie	1	(3)
Linear Lie algebras	4	(3)
Structure constants	7	(3)
Lie algebras from p-groups	10	(3)
On algorithms	13	(4)
Centralizers and normalizers	17	(2)
Chains of ideals	19	(2)
Morphisms of Lie algebras	21	(1)
Derivations	22	(2)
(Semi)direct sums	24	(2)
Automorphisms of Lie algebras	26	(1)
Representations of Lie algebras	27	(2)
Restricted Lie algebras	29	(4)
Extension of the ground field	33	(1)
Finding a direct sum decomposition	34	(4)
Notes	38	(1)
On nilpotency and solvability	39	(18)
Engel's theorem	39	(3)
The nilradical	42	(2)
The solvable radical	44	(3)
Lie's theorems	47	(2)
A criterion for solvability	49	(2)
a characterization of the solvable radical	51	(3)
Finding a non-nilpotent element	54	(2)
Notes	56	(1)
Cartan subalgebras	57	(32)
Primary decompositions	58	(6)
Cartan subalgebras	64	(5)
The root space decomposition	69	(3)
Polynomial functions	72	(1)
Conjugacy of Cartan subalgebras	73	(3)
Conjugacy of Cartan subalgebras of solvable Lie algebras	76	(3)
Calculating the nilradical	79	(7)
Notes	86	(3)
Lie algebras with non-degenerate Killing form	89	(54)
Trace forms and the Killing form	90	(1)
Semisimple Lie algebras	91	(1)
Direct sum decomposition	92	(3)
Complete reducibility of representations	95	(4)
All derivations are inner	99	(2)
The Jordan decomposition	101	(3)
Levi's theorem	104	(3)
Existence of a Cartan subalgebra	107	(1)
Facts on roots	108	(8)
Some proofs for modular fields	116	(4)
Splitting and decomposing elements	120	(8)
Direct sum decomposition	128	(2)
Computing a Levi subalgebra	130	(4)
A structure theorem of Cartan subalgebras	134	(2)
Using Cartan subalgebras to compute Levi subalgebras	136	(5)
Notes	141	(2)
The classification of the simple Lie algebras	143	(76)
Representations of sl2(F)	144	(2)
Some more root facts	146	(4)
Root systems	150	(4)
Root systems of rank two	154	(3)
Simple systems	157	(4)
Cartan matrices	161	(3)
Simple systems and the Weyl group	164	(3)
Dynkin diagrams	167	(2)
Classifying Dynkin diagrams	169	(7)
Constructing the root systems	176	(5)
Constructing isomorphisms	181	(6)
Constructing the semisimple Lie algebras	187	(2)
The simply-laced case	189	(5)
Diagram automorphisms	194	(1)
The non simply-laced case	195	(11)
The classification theorem	206	(1)
Recognizing a semisimple Lie algebra	206	(11)
Identifying a semisimple Lie algebra	207	(6)
Isomorphism of semisimple Lie algebras	213	(4)
Notes	217	(2)
Universal enveloping algebras	219	(38)
Ideals in free associative algebras	220	(6)
Universal enveloping algebras	226	(3)
Grobner bases in universal enveloping algebras	229	(9)
Grobner bases of left ideals	238	(2)
Constructing a representation of a Lie algebra of characteristic 0	240	(13)
Calculating a series of extensions	240	(2)
The extension space	242	(3)
Extending a representation	245	(4)
Ado's theorem	249	(4)
The theorem of Iwasawa	253	(2)
Notes	255	(2)
Finitely presented Lie algebras	257	(54)
Free Lie algebras	258	(3)
Finitely presented Lie algebras	261	(1)
Grobner bases in free algebras	261	(5)
Constructing a basis of a finitely presented Lie algebra	266	(5)
Hall sets	271	(2)
Standard sequences	273	(3)
A Hall set provides a basis	276	(5)
Two examples of Hall orders	281	(4)
Reduction in L(X)	285	(7)
Grobner bases in free Lie algebras	292	(8)
Presentations of the simple Lie algebras of characteristic zero	300	(8)
Notes	308	(3)
Representations of semisimple Lie algebras	311	(52)
The weights of a representation	312	(3)
Verma modules	315	(2)
Integral functions and the Weyl group	317	(4)
Finite dimensionality	321	(2)
On representing the weights	323	(3)
Computing orbits of the Weyl group	326	(5)
Calculating the weights	331	(2)
The multiplicity formula of Freudenthal	333	(5)
Modifying Freudenthal's formula	338	(3)
Weyl's formulas	341	(8)
The formulas of Kostant and Racah	349	(3)
Decomposing a tensor product	352	(7)
Branching rules	359	(2)
Notes	361	(2)
A On associative algebras	363	(16)
A.1 Radical and semisimplicity	364	(3)
A.2 Algebras generated by a single element	367	(6)
A.3 Central idempotents	373	(4)
A.4 Notes	377	(2)
Bibliography	379	(8)
Index of Symbols	387	(2)
Index of Terminology	389	(4)
Index of Algorithms	393