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Almost Commuting Elements in Compact Lie Groups
Armand Borel, Institute for Advanced Study, Princeton, NJ, and Robert Friedman and John W. Morgan, Columbia University, New York City, NY
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Memoirs of the American Mathematical Society
2002; 136 pp; softcover
Volume: 157
ISBN-10: 0-8218-2792-8
ISBN-13: 978-0-8218-2792-5
List Price: US$61 Individual Members: US$36.60
Institutional Members: US\$48.80
Order Code: MEMO/157/747

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.

• Introduction
• Almost commuting $$N$$-tuples
• Some characterizations of groups of type $$A$$
• $$c$$-pairs
• Commuting triples
• Some results on diagram automorphisms and associated root systems
• The fixed subgroup of an automorphism
• $$C$$-triples
• The tori $$\overline{S}(k)$$ and $$\overline{S}^{w_c}(\overline{\bf g}, k)$$ and their Weyl groups
• The Chern-Simons invariant
• The case when $$\langle C\rangle$$ is not cyclic
• Bibliography
• Diagrams and tables