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Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center. (English) Zbl 1107.81044
Summary: The \(\text{SL}(2,\mathbb Z)\)-representation \(\pi\) on the center of the restricted quantum group \(\overline{\mathcal U}s\ell(2)\) at the primitive \(2p\)th root of unity is shown to be equivalent to the \(\text{SL}(2,\mathbb Z)\)-representation on the extended characters of the logarithmic \((1,p)\) conformal field theory model. The multiplicative Jordan decomposition of the \(\overline{\mathcal U}s\ell(2)\) ribbon element determines the decomposition of \(\pi\) into a “pointwise” product of two commuting \(\text{SL}(2,\mathbb Z)\)-representations, one of which restricts to the Grothendieck ring; this restriction is equivalent to the \(\text{SL}(2,\mathbb Z)\)-representation on the \((1,p)\)-characters, related to the fusion algebra via a nonsemisimple Verlinde formula. The Grothendieck ring of \(\overline{\mathcal U}s\ell(2)\) at the primitive \(2 p\)th root of unity is shown to coincide with the fusion algebra of the \((1,p)\) logarithmic conformal field theory model. As a by-product, we derive \(q\)-binomial identities implied by the fusion algebra realized in the center of \(\overline{\mathcal U}s\ell(2)\).

MSC:
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
17B37 Quantum groups (quantized enveloping algebras) and related deformations
20G05 Representation theory for linear algebraic groups
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