Vogel's universality and the classification problem for Jacobi identities

TL;DR

This paper explores Vogel's universality in relation to Lie algebra classification, knot invariants, and potential breakdowns under deformations, with implications for algebraic and physical theories.

Contribution

It provides a summary of Vogel theory's connections to Lie algebra classification, knot invariants, and discusses potential universality breakdowns under deformations.

Findings

Relation between Vogel divisors and Dynkin classification

Application of Jacobi identities to Kontsevich integral

Discussion on universality breakdown after Jack/Macdonald deformation

Abstract

This paper is a summary of discussions at the recent ITEP-JINR-YerPhI workshop on Vogel theory in Dubna. We consider relation between Vogel divisor(s) and the old Dynkin classification of simple Lie algebras. We consider application to knot theory and the hidden role of Jacobi identities in the definition/invariance of Kontsevich integral, which is the knot polynomial with the values in diagrams, capable of revealing all Vassiliev invariants -- including the ones, not visible in other approaches. Finally we comment on the possible breakdown of Vogel universality after the Jack/Macdonald deformation. Generalizations to affine, Yangian and DIM algebras are also mentioned. Especially interesting could be the search for universality in ordinary Yang-Mills theory and its interference with confinement phenomena.