The SL(2, Z)-representation π on the center of the restricted quantum group U q sℓ(2) at the primitive 2pth root of unity is shown to be equivalent to the SL(2, Z)representation on the extended characters of the logarithmic (1, p) conformal field theory model. The multiplicative Jordan decomposition of the U q sℓ(2) ribbon element determines the decomposition of π into a "pointwise" product of two commuting SL(2, Z)representations, one of which restricts to the Grothendieck ring; this restriction is equivalent to the SL(2, Z)-representation on the (1, p)-characters, related to the fusion algebra via a nonsemisimple Verlinde formula. The Grothendieck ring of U q sℓ(2) at the primitive 2pth 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 U q sℓ(2).

ABSTRACT. We study logarithmic conformal field models that extend the (p, q) Virasoro minimal models. For coprime positive integers p and q, the model is defined as the kernel of the two minimal-model screening operators. We identify the field content, construct the W -algebra W p,q that is the model symmetry (the maximal local algebra in the kernel), describe its irreducible modules, and find their characters. We then derive the SL(2, Z)-representation on the space of torus amplitudes and study its properties. From the action of the screenings, we also identify the quantum group that is Kazhdan-Lusztig-dual to the logarithmic model. CONTENTS

ABSTRACT. We derive and study a quantum group g p,q that is Kazhdan-Lusztig-dual to the W -algebra W p,q of the logarithmic (p, q) conformal field theory model. The algebra W p,q is generated by two currents W + (z) and W − (z) of dimension (2p−1)(2q−1), and the energy-momentum tensor T (z). The two currents generate a vertex-operator ideal R with the property that the quotient W p,q /R is the vertex-operator algebra of the (p, q) Virasoro minimal model. The number (2pq) of irreducible g p,q representations is the same as the number of irreducible W p,q -representations on which R acts nontrivially. We find the center of g p,q and show that the modular group representation on it is equivalent to the modular group representation on the W p,q characters and "pseudocharacters." The factorization of the g p,q ribbon element leads to a factorization of the modular group representation on the center. We also find the g p,q Grothendieck ring, which is presumably the "logarithmic" fusion of the (p, q) model. CONTENTS

The interest in Logarithmic Conformal Field Theories (LCFTs) has been growing over the last few years thanks to recent developments coming from various approaches. A particularly fruitful point of view consists in considering lattice models as regularizations for such quantum field theories. The indecomposability then encountered in the representation theory of the corresponding finite-dimensional associative algebras exactly mimics the Virasoro indecomposable modules expected to arise in the continuum limit. In this paper, we study in detail the so-called Temperley-Lieb (TL) fusion functor introduced in physics by Read and Saleur [Nucl. Phys. B 777, 316 (2007)]. Using quantum group results, we provide rigorous calculations of the fusion of various TL modules. Our results are illustrated by many explicit examples relevant for physics. We discuss how indecomposability arises in the "lattice" fusion and compare the mechanisms involved with similar observations in the corresponding field theory. We also discuss the physical meaning of our lattice fusion rules in terms of indecomposable operator-product expansions of quantum fields.Comment: 54pp, many comments adde