Electronic ISBN:  9781470403409 
Product Code:  MEMO/157/747.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 157; 2002; 136 ppMSC: Primary 22; 17; Secondary 57;
We describe the components of the moduli space of conjugacy classes of commuting pairs and triples of elements in a compact Lie group. This description is in terms of the extended Dynkin diagram of the simply connected cover, together with the coroot integers and the action of the fundamental group. In the case of three commuting elements, we compute ChernSimons invariants associated to the corresponding flat bundles over the threetorus, and verify a conjecture of Witten which reveals a surprising symmetry involving the ChernSimons invariants and the dimensions of the components of the moduli space.
ReadershipGraduate students and research mathematicians interested in topological groups, Lie groups, and nonassociative rings and algebras.

Table of Contents

Chapters

1. Introduction

2. Almost commuting $N$tuples

3. Some characterizations of groups of type $A$

4. $c$pairs

5. Commuting triples

6. Some results on diagram automorphisms and associated root systems

7. The fixed subgroup of an automorphism

8. $C$triples

9. The tori $\bar {S}(k)$ and $\bar {S}^{w_C}(\bar {\mathbf {g}},k)$ and their Weyl groups

10. The ChernSimons invariant

11. The case when $\langle C\rangle $ is not cyclic


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We describe the components of the moduli space of conjugacy classes of commuting pairs and triples of elements in a compact Lie group. This description is in terms of the extended Dynkin diagram of the simply connected cover, together with the coroot integers and the action of the fundamental group. In the case of three commuting elements, we compute ChernSimons invariants associated to the corresponding flat bundles over the threetorus, and verify a conjecture of Witten which reveals a surprising symmetry involving the ChernSimons invariants and the dimensions of the components of the moduli space.
Graduate students and research mathematicians interested in topological groups, Lie groups, and nonassociative rings and algebras.

Chapters

1. Introduction

2. Almost commuting $N$tuples

3. Some characterizations of groups of type $A$

4. $c$pairs

5. Commuting triples

6. Some results on diagram automorphisms and associated root systems

7. The fixed subgroup of an automorphism

8. $C$triples

9. The tori $\bar {S}(k)$ and $\bar {S}^{w_C}(\bar {\mathbf {g}},k)$ and their Weyl groups

10. The ChernSimons invariant

11. The case when $\langle C\rangle $ is not cyclic