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Modular transformations of SU(N) affine characters and their commutant. (English) Zbl 0703.17017

The authors examine the action of the modular group on the characters of integrable representations of the Kac-Moody algebra associated to SU(N). Let S and T denote the generators of the modular group, a basis of the algebra of matrices commuting with S and T is described by orbits under the modular action on \(G_ n\times G_ n\), where \(G_ n\) denotes a quotient of the weight lattice of SU(N). It is proved that there is a basis consisting of matrices with integral entries, i.e. a basis of the subalgebra over the rationals of the commutant of S and T. The case \(N=3\) is discussed in some detail by use of an interpretation of the weight lattice as a quadratic field.
Reviewer: H.Boseck

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
81T25 Quantum field theory on lattices
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
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References:

[1] Kac, V.G.: Infinite dimensional Lie algebras, 2nd edn., Cambridge: Cambridge University Press 1985 · Zbl 0574.17010
[2] Cappelli, A., Itzykson, C., Zuber, J.-B.: The A-D-E classification of minimal andA 1 (1) conformal invariant theories. Commun. Math. Phys.113, 1, 26 (1987) · Zbl 0639.17008 · doi:10.1007/BF01221394
[3] Curtis, C.W., Rainer, I.: Representation theory of finite groups and associative algebras, Chap. 1. New York: Wiley 1988
[4] Christe, P., Ravanini, F.:G NL/GL+N conformal field theories and their modular invariant partition functions. Int. J. Mod. Phys. A4, 897–920 (1989) · Zbl 0696.17013 · doi:10.1142/S0217751X89000418
[5] Moore, G., Seiberg, N.: Naturality in conformal field theory. Nucl. Phys. B313, 16–40 (1989) · Zbl 0694.53074 · doi:10.1016/0550-3213(89)90511-7
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