$X_2$ series of universal quantum dimensions

TL;DR

This paper derives universal formulas for quantum dimensions of Cartan powers of the $X_2$ representation in simple Lie algebras, with applications to universal knot polynomials.

Contribution

It provides the first universal formulas for quantum dimensions of $X_2$, expanding understanding of Lie algebra representations and their role in knot theory.

Findings

Universal quantum dimension formulas for $X_2$ representations.

Analysis of singular cases and Vogel's parameters.

Application to universal knot polynomials.

Abstract

The antisymmetric square of the adjoint representation of any simple Lie algebra is equal to the sum of adjoint and $X_2$ representations. We present universal formulae for quantum dimensions of an arbitrary Cartan power of $X_2$. They are analyzed for singular cases and permuted universal Vogel's parameters. $X_2$ has been the only representation in the decomposition of the square of the adjoint with unknown universal series. Application to universal knot polynomials is discussed.