Arithmetic of universality in simple Lie algebras

We introduce notion of universality in simple Lie algebras and their applications, on the basis of works of Vogel and Deligne. We briefly review different aspects of universality: universal formulae for dimensions of representations, eigenvalues of higher Casimir operators, universal characters, characters’ hypothesis, etc. We define "population" of Vogel's plane as points for which universal character of adjoint representation is regular in the finite plane of its argument, as it is the case for simple Lie algebras. We show that they are given exactly by all solutions of seven Diophantine equations of third order on three variables. One of these Diophantine equations, namely knm=4k+4n+2m+12, contains all simple Lie algebras, except so(2N+1). We list all solutions and show that they contain all simple Lie (super)algebras as well as few other points with interesting properties.