Baby bead representations

TL;DR

This paper investigates baby bead representations through Lie algebra homology, employing truncation techniques and auxiliary categories to derive explicit results and analyze composition factors of bead representations.

Contribution

It introduces a novel approach using truncation and auxiliary categories to analyze bead representations, providing explicit results and new insights into their composition factors.

Findings

Complete results for baby bead representations are obtained.

The composition factors of bead representations are calculated for a new infinite family.

The approach reveals non-trivial information despite truncation.

Abstract

This paper is motivated by the study of Turchin and Willwacher's bead representations. The problem is reformulated here in terms of the Lie algebra homology of a free Lie algebra with coefficients in tensor products of the adjoint representation. The main idea is to exploit the truncation of the coefficients given by killing Lie brackets of length greater than two. Although this truncation is brutal, it retains significant and highly non-trivial information, as exhibited by explicit results. A d\'evissage is used that splits the problem into two steps, separating out a `homology' calculation from `antisymmetrization'. This involves some auxiliary categories, including a generalization of the upper walled Brauer category. This approach passes through the `baby bead representations' of the title, for which complete results are obtained. As an application, the composition factors of Turchin and Willwacher's bead representations are calculated for a new infinite family.