Vogel universality for representations ad$^k$ for $k = 2, 3, 4, 5$ and colour factors of Feynman diagrams

In the late 20th century, P. Vogel proposed the concept of a universal Lie algebra to describe properties common to all simple Lie algebras and certain Lie superalgebras, specifically concerning the tensor powers of their adjoint representations. This universality means that many characteristics of these algebras are expressed by rational homogeneous functions of three Vogel parameters, which take fixed values for each specific algebra. While Vogel’s original derivation stemmed from knot theory, this work proposes an alternative approach based on the so-called split Casimir operator. This method yields a constructive algorithm for the universal decomposition of tensor powers of the adjoint representation into irreducible components, significantly simplifying the study of their universal properties. Results for the square, cube, fourth power, and antisymmetric part of the fifth power of the adjoint representation are presented.
Furthermore, the split Casimir operator has an interpretation in terms of color factors for Feynman diagrams. This connection enables the derivation of universal expressions for some of these color factors, which will also be discussed.