Cellularity of certain quantum endomorphism algebras

Henning H. Andersen, Gustav I. Lehrer and Ruibin Zhang

Vol. 279 (2015), No. 1-2, 11–35

DOI: 10.2140/pjm.2015.279.11

Abstract

For any ring $\tilde{A}$ such that $\mathbb{Z}\left[q^{\pm 1∕2}\right]\subseteq \tilde{A}\subseteq \mathbb{Q}\left(q^{1∕2}\right)$, let ${\Delta }_{\tilde{A}}\left(d\right)$ be an $\tilde{A}$-form of the Weyl module of highest weight $d\in \mathbb{N}$ of the quantised enveloping algebra $U_{\tilde{A}}$ of ${\mathfrak{s}\mathfrak{l}}_2$. For suitable $\tilde{A}$, we exhibit for all positive integers $r$ an explicit cellular structure for ${End}_{U_{\tilde{A}}}\left({\Delta }_{\tilde{A}}{\left(d\right)}^{\otimes r}\right)$. This algebra and its cellular structure are described in terms of certain Temperley–Lieb-like diagrams. We also prove general results that relate endomorphism algebras of specialisations to specialisations of the endomorphism algebras. When $\zeta$ is a root of unity of order bigger than $d$ we consider the $U_{\zeta }$-module structure of the specialisation ${\Delta }_{\zeta }{\left(d\right)}^{\otimes r}$ at $q\mapsto \zeta$ of ${\Delta }_{\tilde{A}}{\left(d\right)}^{\otimes r}$. As an application of these results, we prove that knowledge of the dimensions of the simple modules of the specialised cellular algebra above is equivalent to knowledge of the weight multiplicities of the tilting modules for $U_{\zeta }\left({\mathfrak{s}\mathfrak{l}}_2\right)$. As an example, in the final section we independently recover the weight multiplicities of indecomposable tilting modules for $U_{\zeta }\left({\mathfrak{s}\mathfrak{l}}_2\right)$ from the decomposition numbers of the endomorphism algebras, which are known through cellular theory.

Keywords

quantum invariants, cellular algebras, tilting modules

Mathematical Subject Classification 2010

Primary: 17B37, 20G42

Secondary: 81R50

Milestones

Received: 3 July 2014

Revised: 25 October 2014

Accepted: 25 October 2014

Published: 21 December 2015