The "Vogel plane" is the quotient of the two-dimensional complex projective space \(P^2\) by the action of the group \(S_3\) of permutations on homogeneous coordinates. The Vogel plane provides a coordinatisation of all simple Lie algebras. A universal function (which "universalises" some quantity for Lie algebras) is a function on the Vogel plane such that the value of this function at the point of the Vogel plane corresponding to a given simple Lie algebra is equal to the value of this quantity on this simple Lie algebra. A "universal formula" is an identification of a quantity defined for simple Lie algebras with a universal function (if possible).
Beginning from the works of Vogel, universal functions for different quantities were discovered. Examples include: dimensions of simple Lie algebras; dimensions of (some) irreducible components of powers of the adjoint representations; eigenvalues of Casimir operators on adjoint representations; partition function of the Chern--Simons theory on the \(3\)-dimensional sphere, etc.
We shall discuss the recently obtained universal function for volumes of compact Lie groups corresponding to simple Lie algebras. It is a "universalisation" of the Kac-Peterson formula. This universal formula expresses volume in terms of Barnes functions. The essential moment is that this formula defines a piecewise-analytic function on the Vogel plane. We perform an analytical continuation of this function and we come to multivalued function. For obtained function we calculate its transformation (in the physical language, "anomaly") under a permutation of Vogel's parameters. This is a universal generalisation of relations which describes asymmetry for the volume formula of \(SU(N)\) under \(N\to -N\) transformations.
In the second part of the talk, we consider Diophantine equation, \(knm=2kn+2km+2nm\), that arises for simple Lie algebras from some remarkable universal formula, and discuss the fact that the same equations appear in the classification of Platonic solids.
The talk is based on recent works of R. L. Mkrtchyan and the speaker:
1. Universal volume of groups and anomaly of Vogel's symmetry.
2. Diophantine equations, Platonic solids, McKay correspondence, equivelar maps and Vogel's universality.