The partial Temperley-Lieb algebra and its representations

We give a combinatorial description of a new diagram algebra, the partial Temperley–Lieb algebra, arising as the generic centralizer algebra ${\mathrm{E}\mathrm{n}\mathrm{d}}_{{\mathbf{U}}_q({\mathfrak{g}\mathfrak{l}}_2)}(V^{\otimes k})$, where $V=V(0)\oplus V(1)$ is the direct sum of the trivial and natural module for the quantized enveloping algebra ${\mathbf{U}}_q({\mathfrak{g}\mathfrak{l}}_2)$. It is a proper subalgebra of the Motzkin algebra (the ${\mathbf{U}}_q({\mathfrak{s}\mathfrak{l}}_2)$-centralizer) of Benkart and Halverson. We prove a version of Schur–Weyl duality for the new algebras, and describe their generic representation theory.