  Let $k$ be an arbitrary field and let $q \in k\setminus\{0\}$. In this paper
we use the known tilting theory for the quantum group $U_q(sl_2)$ to obtain the
dimensions of simple modules for the Temperley-Lieb algebras $TL_n(q+q^{-1})$
and related algebras over $k$. Our main result is an algorithm which calculates
the dimensions of simple modules for these algebras. We take advantage of the
fact that $TL_n(q+q^{-1})$ is isomorphic to the endomorphism ring of the $n$'th
tensor power of the natural $2$-dimensional module for the quantum group for
$sl_2$. This algorithm is easy when the characteristic is $0$ and more involved
in positive characteristic. We point out that our results for the
Temperley-Lieb algebras contain a complete description of the simple modules
for the Jones quotient algebras. Moreover, we illustrate how the same results
lead to corresponding information about simple modules for the BMW-algebras and
other algebras closely related with endomorphism algebras of families of
tilting modules for $U_q(sl_2)$.
