The partial Temperley–Lieb algebra and its representations

Abstract

In this paper, we give a combinatorial description of a new diagram algebra, the partial Temperley–Lieb algebra, arising as the generic centralizer algebra \mathrm{End}_{\mathbf{U}_q(\mathfrak{gl}_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{gl}_2) . It is a proper subalgebra of the Motzkin algebra (the \mathbf{U}_q(\mathfrak{sl}_2) -centralizer) of Benkart and Halverson. We prove a version of Schur–Weyl duality for the new algebras, and describe their generic representation theory.