  We exhibit for all positive integers r, an explicit cellular structure for
the endomorphism algebra of the r'th tensor power of an integral form of the
Weyl module with highest weight d of the quantised enveloping algebra of sl2.
When q is specialised to a root of unity of order bigger than d, we consider
the corresponding specialisation of the tensor power. We prove one general
result which gives sufficient conditions for the commutativity of
specialisation with the taking of endomorphism algebras, and another which
relates the multiplicities of indecomposable summands to the dimensions of
simple modules for an endomorphism algebra. Our cellularity result then allows
us to prove that knowledge of the dimensions of the simple modules of the
specialised cellular algebra above is equivalent to knowledge of the weight
multiplicities of the tilting modules for the specialised quantum group. In the
final section we independently determine the weight multiplicities of
indecomposable tilting modules for quantum sl2, and the decomposition numbers
of the endomorphism algebras. We indicate how either one of these sets of
numbers determines the other.
