  We investigate a subalgebra of the Temperley-Lieb algebra called the
Jones-Wenzl algebra, which is obtained by action of certain Jones-Wenzl
projectors. This algebra arises naturally in applications to conformal field
theory and statistical physics. It is also the commutant (centralizer) algebra
of the Hopf algebra $U_q(\mathfrak{sl}_2)$ on its type-one modules - this fact
is a generalization of the $q$-Schur-Weyl duality of Jimbo. In this article, we
find two minimal generating sets for the Jones-Wenzl algebra. In special cases,
we also find all of the independent relations satisfied by these generators.
