Diagrammatic technique for Vogel's universality

TL;DR

This paper revisits Vogel's diagrammatic algebra framework to perform universal computations in Lie theory, demonstrating its effectiveness through various examples.

Contribution

It revitalizes Vogel's diagrammatic approach, showing it enables universal Lie algebra calculations that complement existing representation-theoretic formulas.

Findings

Diagrammatic technique enables universal Lie algebra computations.

Many examples demonstrate the effectiveness of the approach.

Revives interest in Vogel's original algebraic framework.

Abstract

In his 1999 preprint "Universal Lie Algebra", P. Vogel put forward a hypothesis on the existence of a universal Lie algebra. Although this hypothesis remains open, it is known that many quantities in Lie theory admit universal descriptions. Remarkably, almost all such universal formulas have been obtained through the representation theory of simple Lie (super)algebras, whereas Vogel's original framework was based on a more abstract diagrammatic algebra. Nevertheless, the diagrammatic approach has received little attention over the past two decades, since the last contributions by P. Vogel and J. Kneissler. In this work, we revive the diagrammatic technique grounded in Vogel's $\Lambda$-algebra and show that it enables truly universal computations. We examine numerous examples and discuss them.