Lie algebras in symmetric monoidal categories

TL;DR

This paper explores Lie algebras within symmetric monoidal categories, establishing a universal connection between Vogel's Lie algebra and Rozansky-Witten invariants of K3 surfaces, with broader implications for knot theory and invariants.

Contribution

It proves Westbury's conjecture linking Vogel's universal Lie algebra to Rozansky-Witten invariants of K3 surfaces, expanding the understanding of Lie algebras in categorical contexts.

Findings

Proof of Westbury's conjecture for K3 surfaces

Establishment of a universal Lie algebra homomorphism

Formulation of nine open research questions

Abstract

We study algebras defined by identities in symmetric monoidal categories. Our focus is on Lie algebras. Besides usual Lie algebras, there are examples appearing in the study of knot invariants and Rozansky-Witten invariants. Our main result is a proof of Westbury's conjecture for K3-surface: there exists a Lie algebra homomorphism from Vogel's universal simple Lie algebra to the Lie algebra describing the Rozansky-Witten invariants of a K3-surface. Most of the paper involves setting up a proper language to discuss the problem and we formulate nine open questions as we proceed.