For any ring
$\tilde{A}$
such that
$\mathbb{Z}\left[{q}^{\pm 1\u22152}\right]\subseteq \tilde{A}\subseteq \mathbb{Q}\left({q}^{1\u22152}\right)$,
let
${\Delta}_{\tilde{A}}\left(d\right)$ be an
$\tilde{A}$form of the Weyl module
of highest weight
$d\in \mathbb{N}$ of the
quantised enveloping algebra
${U}_{\tilde{A}}$
of
${\mathfrak{s}\mathfrak{l}}_{2}$. For suitable
$\tilde{A}$, we exhibit for all positive
integers
$r$ an explicit
cellular structure for
${End}_{{U}_{\tilde{A}}}\left({\Delta}_{\tilde{A}}{\left(d\right)}^{\otimes r}\right)$.
This algebra and its cellular structure are described in terms of certain
Temperley–Lieblike diagrams. We also prove general results that relate endomorphism
algebras of specialisations to specialisations of the endomorphism algebras. When
$\zeta $ is a root of unity of order
bigger than
$d$ we consider
the
${U}_{\zeta}$module structure
of the specialisation
${\Delta}_{\zeta}{\left(d\right)}^{\otimes r}$
at
$q\mapsto \zeta $ of
${\Delta}_{\tilde{A}}{\left(d\right)}^{\otimes r}$. As
an application of these results, we 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
${U}_{\zeta}\left({\mathfrak{s}\mathfrak{l}}_{2}\right)$. As an example,
in the final section we independently recover the weight multiplicities of indecomposable tilting
modules for
${U}_{\zeta}\left({\mathfrak{s}\mathfrak{l}}_{2}\right)$
from the decomposition numbers of the endomorphism algebras, which are known
through cellular theory.
