The partial Temperley-Lieb algebra and its representations

TL;DR

This paper introduces the partial Temperley-Lieb algebra, a new diagram algebra arising as a centralizer in quantum group representations, and explores its structure, duality, and representation theory.

Contribution

It provides a combinatorial description of the partial Temperley-Lieb algebra and establishes Schur--Weyl duality for it, expanding understanding of quantum group centralizers.

Findings

Defined the partial Temperley-Lieb algebra as a subalgebra of the Motzkin algebra.

Proved a version of Schur--Weyl duality for the new algebra.

Described the generic representation theory of the algebra.

Abstract

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.