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

The SL(2, Z)-representation pi on the center of the restricted quantum group (U) over bar (q)sl(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 (U) over bar (q)sl(2) ribbon element determines the decomposition of pi 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) over bar (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) over bar (q)sl(2).