Memoirs of the American Mathematical Society 2002; 136 pp; softcover Volume: 157 ISBN10: 0821827928 ISBN13: 9780821827925 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 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. Readership Graduate students and research mathematicians interested in topological groups, Lie groups, and nonassociative rings and algebras. Table of Contents  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 ChernSimons invariant
 The case when \(\langle C\rangle\) is not cyclic
 Bibliography
 Diagrams and tables
