Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center

The S L (2, ℤ)-representation π on the center of the restricted quantum group Ū_qsℓ(2) at the primitive 2p^th 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 Ū_qsℓ(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 Ū_qsℓ(2) at the primitive 2p^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 Ū_qsℓ(2).