Lie Algebras: Theory and Algorithms (eBook)

W.A. de Graaf (Autor)

eBook Download: PDF

2000 | 1. Auflage
408 Seiten
Elsevier Science (Verlag)
978-0-08-053545-6 (ISBN)

Infos

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.

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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 Poincare-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.

Inhalt

Front Cover 1
Lie Algebras: Theory and Algorithms 4
Copyright Page 5
Contents 10
Chapter 1. Basic constructions 14
1.1 Algebras: associative and Lie 14
1.2 Linear Lie algebras 17
1.3 Structure constants 20
1.4 Lie algebras from p-groups 23
1.5 On algorithms 26
1.6 Centralizers and normalizers 30
1.7 Chains of ideals 32
1.8 Morphisms of Lie algebras 34
1.9 Derivations 35
1.10 (Semi)direct sums 37
1.11 Automorphisms of Lie algebras 39
1.12 Representations of Lie algebras 40
1.13 Restricted Lie algebras 42
1.14 Extension of the ground field 46
1.15 Finding a direct sum decomposition 47
1.16 Notes 51
Chapter 2. On nilpotency and solvability 52
2.1 Engel’s theorem 52
2.2 The nilradical 55
2.3 The solvable radical 57
2.4 Lie’s theorems 60
2.5 A criterion for solvability 62
2.6 A characterization of the solvable radical 64
2.7 Finding a non-nilpotent element 67
2.8 Notes 69
Chapter 3. Cartan subalgebras 70
3.1 Primary decompositions 71
3.2 Cartan subalgebras 77
3.3 The root space decomposition 82
3.4 Polynomial functions 85
3.5 Conjugacy of Cartan subalgebras 86
3.6 Conjugacy of Cartan subalgebras of solvable Lie algebras 89
3.7 Calculating the nilradical 92
3.8 Notes 99
Chapter 4. Lie algebras with non-degenerate Killing form 102
4.1 Trace forms and the Killing form 103
4.2 Semisimple Lie algebras 104
4.3 Direct sum decomposition 105
4.4 Complete reducibility of representations 108
4.5 All derivations are inner 112
4.6 The Jordan decomposition 114
4.7 Levi’s theorem 117
4.8 Existence of a Cartan subalgebra 120
4.9 Facts on roots 121
4.10 Some proofs for modular fields 129
4.11 Splitting and decomposing elements 133
4.12 Direct sum decomposition 141
4.13 Computing a Levi subalgebra 143
4.14 A structure theorem of Cartan subalgebras 147
4.15 Using Cartan subalgebras to compute Levi subalgebras 149
4.16 Notes 154
Chapter 5. The classification of the simple Lie algebras 156
5.1 Representations of sI2(F) 157
5.2 Some more root facts 159
5.3 Root systems 163
5.4 Root systems of rank two 167
5.5 Simple systems 170
5.6 Cartan matrices 174
5.7 Simple systems and the Weyl group 177
5.8 Dynkin diagrams 180
5.9 Classifying Dynkin diagrams 182
5.10 Constructing the root systems 189
5.11 Constructing isomorphisms 194
5.12 Constructing the semisimple Lie algebras 200
5.13 The simply-laced case 202
5.14 Diagram automorphisms 207
5.15 The non simply-laced case 208
5.16 Thc classification theorem 219
5.17 Recognizing a semisimple Lie algebra 219
5.18 Notes 230
Chapter 6. Universal enveloping algebras 232
6.1 Ideals in free associative algebras 233
6.2 Universal enveloping algebras 239
6.3 Gröbner bases in universal enveloping algebras 242
6.4 Gröbner bases of left ideals 251
6.5 Constructing a representation of a Lie algebra of characteristic 0 253
6.6 The theorem of Iwasawa 266
6.7 Notes 268
Chapter 7. Finitely presented Lie algebras 270
7.1 Free Lie algebras 271
7.2 Finitely presented Lie algebras 274
7.3 Gröbner bases in free algebras 274
7.4 Constructing a basis of a finitely presented Lie algebra 279
7.5 Hall sets 284
7.6 Standard sequences 286
7.7 A Hall set provides a basis 289
7.8 Two examples of Hall orders 294
7.9 Reduction in L(X) 298
7.10 Gröbner bases in free Lie algebras 305
7.11 Presentations of the simple Lie algebras of characteristic zero 313
7.12 Notes 321
Chapter 8. Representations of semisimple Lie algebras 324
8.1 The weights of a representation 325
8.2 Verma modules 328
8.3 Integral functions and the Weyl group 330
8.4 Finite dimensionality 334
8.5 On representing the weights 336
8.6 Computing orbits of the Weyl group 339
8.7 Calculating the weights 344
8.8 The multiplicity formula of Freudenthal 346
8.9 Modifying Freudenthal’s formula 351
8.10 Weyl’s formulas 354
8.11 The formulas of Kostant and Racah 362
8.12 Decomposing a tensor product 365
8.13 Branching rules 372
8.14 Notes 374
Appendix A. On associative algebras 376
A.1 Radical and semisimplicity 377
A.2 Algebras generated by a single element 380
A.3 Central idempotents 386
A.4 Notes 390
Bibliography 392
Index of Symbols 400
Index of Terminology 402
Index of Algorithms 406

Textauszug

1.10 (Semi)direct sums

Let L1 and L2 be Lie algebras over the field F. In this section we describe how we can define a multiplication on the direct sum (of vector spaces) L1 ⊕ L2 extending the multiplications on L1,L2 and making L1 ⊕ L2 into a Lie algebra. Let the product on L/ be denoted by [,]1 and on L2 by [,]2∙ Suppose we have a morphism of Lie algebras θ: L1 → Der(L2) · Then this map allows us to define an algebra structure on the vector space L1 ⊕ L2 by setting

1+x2,y1+y2=x1y11+θx1y2−θy1x2+x2y22

  (1.12)

for x1, y1 ∈ L1 and x2, y2 ∈ L2. (So the product [x, y] for x ∈ L1 and y ∈ L2 is formed by applying the derivation θ(x) to y.)

Lemma 1.10.1

The multiplication defined by (1–12) makes L1 ⊕ L2 into a Lie algebra.

Proof. Let x1 ∈ L1 and x2 ∈ L2. Then

1+x2,x1+x2=x1x11+θx1x2−θx1x2+x2x22=0.

It suffices to check the Jacobi identity for the elements of a basis of the Lie algebra (cf. Lemma 1.3.1). Construct a basis of L1 ⊕ L2 by first taking a basis of L1 and adding a basis of L2. Let x,y,z be three elements from this basis. We have to consider a few cases: all three elements are from L1, two elements are from L1 (the other from L2), one element is from L1, no elements are from L1. The first and the last case are implied by the Jacobi identity on L1 and L2 respectively. The second case follows from the fact that θ is a morphism of Lie algebras. And the third case follows from the fact that θ(x) is a derivation of L2 for x ∈ L1. (We leave the precise verifications to the reader).

The Lie algebra L1 ⊕ L2, together with the multiplication defined by (1.12), is called the semidirect sum of L1 and L2 (with respect to θ). It is straightforward to see that it contains L2 as an ideal and L1 as a subalgebra.

On the other hand, suppose that a Lie algebra L contains a subalgebra L1 and an ideal L2 such that as a vector space L = L1 ⊕ L2. Then for x ∈ L1 and y ∈ L2 we have adx(y) = [x,y] ∈ L2 because L2 is an ideal. As a consequence adx maps L2 into itself. Furthermore, as seen in Example 1.9.1, the map adx is a derivation of L2 Hence we have a map dL2:L1→DerL2. By Example 1.8.3, dL2 is a morphism of Lie algebras. It is immediate that the product on L satisfies (1.12) with =adL2 So L is the semidirect sum of L1 and L2.

If there can be no confusion about the map θ, then the semidirect sum of L1 and L2 is denoted by L1 × L2.

A special case of the construction of the semidirect sum occurs when we take the map θ to be identically 0. Then the multiplication on L1 ⊕ L2 is given by

1+x2,y1+y2=x1y11+x2y22forallx1,y1∈L1andx2,y2∈L2.

The Lie algebra L1 ⊕ L2 together with this product is called the direct sum (of Lie algebras) of L1 and L2. It is simply denoted by L1 ⊕ L2. From the way in which the product on L1 ⊕ L2 is defined, it follows that L1 and L2 are ideals in L1 ⊕ L2. Conversely suppose that L is a Lie algebra that is the direct sum of two subspaces L1 and L2. Furthermore, suppose that L1 and L2 happen to be ideals of L. Then [L1,L2] is contained in both L1 and L2. Therefore [L1,L2] = 0 and it follows that L is the direct sum (of Lie algebras) of L1 and L2.

Example 1.10.2

In general a Lie algebra can be a semidirect sum in more than one way. Let L1 and L2 be copies of the same Lie algebra L. Let ϕ:L1 → L2 be the algebra morphism induced by the identity on L. Define θ: L1 → Der(L2) by x=adL2ϕx for x ∈ L1. Then θ is the composition of two morphisms of Lie algebras. Consequently, θ is a morphism of Lie algebras and hence we can form the semidirect sum of L1 and L2 with respect to θ. Denote the resulting Lie algebra by K. Now let K1 be the subspace of K spanned by all elements of the form x – ϕ(x) for x ∈ L1. And let K2 be the subspace spanned by all ϕ(x) for x ∈ L1 (i.e., K2 is equal to L2). Then for x,y ∈ L1,

−ϕx,y−ϕy=xy−θxϕy+θyϕx+ϕx,ϕy=xy−ϕx,ϕy+ϕy,ϕx+ϕx,ϕy=xy−ϕx,ϕy=xy−ϕxy.

Hence K1 is a subalgebra of K. By a similar calculation it can be shown that [K1,K2] = 0 so that K1 and K2 are ideals of K. So K is the direct sum (of Lie algebras) of K1 and K2.

In Section 1.15 we will give an algorithm for finding a decomposition of L as a direct sum of ideals.

1.11 Automorphisms of Lie algebras

Let L be a Lie algebra over the field F. An automorphism of L is an isomorphism of L onto itself. Since products of automorphisms are automorphisms and inverses of automorphisms are automorphisms, the automorphisms of L form a group. This group is called the automorphism group of L; it is denoted by Aut(L).

Example 1.11.1

Let V be a finite-dimensional vector space. Let L be a subalgebra of lV and g ∈ End (V) an invertible endomorphism of V. If gLg−l = L then the map x  gxg− 1 is an automorphism of L because [gxg− 1,gyg− 1] = gxyg− 1−gyxg− 1 = g[x,y]g− 1·

Now let the ground field F be of characteristic 0. If d is a nilpotent derivation of L, i.e., dn = 0 for some integer n ≥ 0, then we can define its exponential:

exp⁡d=1+d+12!d2+…+1n−1!dn−1.

Lemma 1.11.2

Let d be a nilpotent derivation of L, then exp d is an automorphism of L.

Proof. Because d is nilpotent, there is some integer n with dn = 0. Now we calculate

⁡dx,exp⁡dy=∑i=0n−1∑j=0n−11i!j!dix,djy=∑m=02n−2∑i=0m1i!m−i!dix,dm−iy=∑m=02n−2dmxym!by1.11=∑m=0n−1dmxym!=exp⁡dxy.

So exp d is a morphism of Lie algebras.

The inverse of exp d is the map

exp⁡−d=∑j=0n−1−1j1j!dj.

It follows that exp d is a bijective morphism of Lie algebras; i.e., it is an automorphism.

In particular, if adx is nilpotent then exp(adx) is an automorphism of L. An...

Produkt-Details

• Erscheint lt. Verlag: 4.2.2000
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 0-08-053545-3 / 0080535453
• ISBN-13: 978-0-08-053545-6 / 9780080535456