Universal dimensions of simple Lie algebras and configurations of points and lines

Abstract

The research, conducted by Vogel in 1999, in which the tensor category, called universal Lie algebra was introduced, provided a parametrization of the simple Lie algebras by three so-called universal parameters ( α : β : γ ) - projective coordinates in Vogel plane. Subsequently, it has been shown, that several characteristics of simple Lie algebras, such as dimensions of certain representations, can be expressed in terms of these three parameters by some analytic functions, which are called universal formulae. We investigate the uniqueness of the known universal dimension formulae, i.e. the possibility of the derivation of two different functions, yielding the same outputs at the same distinguished points. We employ the recently revealed geometrical rephrasing of this problem, which links us to a completely different area of mathematics - the theory of configurations of points and lines, particularly, we derive an explicit expression for a four-by-four non-uniqueness factor, making use of a known ( 16 3 , 12 4 ) configuration, demonstrating the benefit the geometrical interpretation provides with.