  The SL(2,Z) representation $\pi$ on the center of the restricted quantum
group U_{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_{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_{q}sl(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~U_{q}sl(2).
