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Almost commuting elements in compact Lie groups. (English) Zbl 0993.22002
Mem. Am. Math. Soc. 747, 136 p. (2002).
Let \(G\) be a compact, connected and semisimple Lie group. The authors describe in this book the components of the moduli space of conjugacy classes of commuting pairs and triples of elements in \(G\). 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, the authors 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. It is also useful to give the list of all chapters of this book: Chapter 1. Introduction. Chapter 2. Almost commuting \(N\)-tuples. Chapter 3. Some characterizations of groups of type \(A\). Chapter 4. \(c\)-pairs. Chapter 5. Commuting triples 39. Chapter 6. Some results on diagram automorphisms and associated root systems. Chapter 7. The fixed subgroup of an automorphism. Chapter 8. \(C\)-triples. Chapter 9. The tori \(\bar S(k)\) and \(\bar S^{wc}(\bar g, k)\) and their Weyl groups. Chapter 10. The Chern-Simons invariant. Chapter 11. The case when \(\langle C\rangle\) is not cyclic.

22C05 Compact groups
17B20 Simple, semisimple, reductive (super)algebras
57R20 Characteristic classes and numbers in differential topology
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
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