Split Casimir operator, Vogel parameters, and color factors in gauge theories

At the end of the 20th century, P. Vogel introduced the concept of a universal Lie algebra, which is conjectured to be a model of all simple Lie algebras and some Lie superalgebras. Universality lies in the fact that many quantities characterizing each simple Lie algebra in various representations appearing in tensor powers of the adjoint representation are expressed as rational homogeneous functions of three Vogel parameters, which take specific values for each simple Lie algebra (and Lie superalgebra). In Vogel's original work, the notion of the universal algebra was obtained using arguments originating in knot theory. This talk is devoted to an alternative approach to studying the universal algebra, based on investigating the properties of the so-called split Casimir operator. This approach allows one to construct a relatively simple algorithm for the universal decomposition of tensor powers of the adjoint representation into subrepresentations and significantly simplifies the study of universal properties of Lie algebras and Lie superalgebras. The talk will present the results of studying the universal structure of the tensor square and the tensor cube of the adjoint representation. Furthermore, the split Casimir operator admits an interpretation in terms of color factors of Feynman diagrams. This fact allows one to write down universal expressions for some such color factors, which will also be discussed in the talk.