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Almost Commuting Elements in Compact Lie Groups

Armand Borel Institute for Advanced Study, Princeton, NJ
Robert Friedman Columbia University, New York, NY
John W. Morgan Columbia University, New York City, NY
Available Formats:
Electronic ISBN: 978-1-4704-0340-9
Product Code: MEMO/157/747.E
List Price: $65.00 MAA Member Price:$58.50
AMS Member Price: $39.00 Click above image for expanded view Almost Commuting Elements in Compact Lie Groups Armand Borel Institute for Advanced Study, Princeton, NJ Robert Friedman Columbia University, New York, NY John W. Morgan Columbia University, New York City, NY Available Formats:  Electronic ISBN: 978-1-4704-0340-9 Product Code: MEMO/157/747.E  List Price:$65.00 MAA Member Price: $58.50 AMS Member Price:$39.00
• Book Details

Memoirs of the American Mathematical Society
Volume: 1572002; 136 pp
MSC: 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 Chern-Simons invariants associated to the corresponding flat bundles over the three-torus, and verify a conjecture of Witten which reveals a surprising symmetry involving the Chern-Simons 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 Chern-Simons invariant
• 11. The case when $\langle C\rangle$ is not cyclic
• Requests

Review Copy – for reviewers who would like to review an AMS book
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Volume: 1572002; 136 pp
MSC: 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 Chern-Simons invariants associated to the corresponding flat bundles over the three-torus, and verify a conjecture of Witten which reveals a surprising symmetry involving the Chern-Simons 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 Chern-Simons invariant
• 11. The case when $\langle C\rangle$ is not cyclic
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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