$(\ell,p)$-Jones-Wenzl Idempotents

TL;DR

This paper extends the understanding of Jones-Wenzl idempotents within the Temperley-Lieb algebra over pointed fields, providing a recursive formula for projective covers of trivial modules, broadening their algebraic applicability.

Contribution

It introduces a recursive form for Jones-Wenzl idempotents over pointed fields, generalizing previous results to new algebraic settings.

Findings

Derived a recursive formula for idempotents

Extended existing results to pointed fields

Described projective covers of trivial modules

Abstract

The Jones-Wenzl idempotents of the Temperley-Lieb algebra are celebrated elements defined over characteristic zero and for generic loop parameter. Given pointed field $(R, \delta)$, we extend the existing results of Burrull, Libedinsky and Sentinelli to determine a recursive form for the idempotents describing the projective cover of the trivial ${\rm TL}_n^R(\delta)$-module.