On linear resolvability of universal quantum dimensions

TL;DR

This paper investigates the linear resolvability of universal quantum dimension formulas at special points in Vogel's plane, demonstrating finiteness and regularity properties that connect to Lie algebra structures and classical limits.

Contribution

It proves that universal quantum dimension formulas are linearly resolvable at key points in Vogel's plane, revealing new regularity phenomena and connections to classical Lie algebra dimensions.

Findings

Universal formulas are linearly resolvable at Vogel's table points.

Three additional distinguished points exhibit regularity for all known quantum dimensions.

Some points yield integer-valued dimensions in the classical limit.

Abstract

In his study of finite (Vassiliev's) knot invariants,Vogel introduced the so-called universal parameters, belonging to the projective plane, which particularly parameterize the simple Lie algebras by the Vogel's table. Subsequently a number of quantities, such as some universal knot invariants, (quantum) dimensions of simple Lie algebras, etc., have been represented in terms of these parameters, i.e. in the universal form. We prove that at the points from the Vogel's table all known universal quantum dimension formulae are linearly resolvable, i.e. yield finite answers even if these points are singular, provided one restricts them to the appropriate lines. We show, that the same phenomenon takes place for another three distinguished points in Vogel's plane - $E_{7\frac{1}{2}}, X_1,$ and $X_2$. We also examine the same formulae on linear resolvability at the remaining 48 distinguished points in Vogel's plane, which correspond to the so-called $Y$-objects, and discover that there are three points among them, which are regular for all known quantum dimension formulae. Two of them happen to be sharing a remarkable similarity with the simple Lie algebras, namely, the universal formulae yield integer-valued outputs (dimensions) at those points in the classical limit.