On the map of Vogel's plane

Abstract

We search points in a Vogel plane with regular universal expression for character of adjoint representation. This gives seven patterns of singularities cancellation, each giving a certain Diophantine equation of third order on three variables. Solutions of these equations are classical series of simple Lie algebras (including an "odd symplectic" one), $D_{2,1,\lambda}$ superalgebra, the straight line of three-dimensional algebras, and a number of isolated solutions, including exceptional simple Lie algebras. One of these Diophantine equations, namely knm=4k+4n+2m+12 contains all simple Lie algebras, except SO(2N+1). Isolated solutions contain, beside exceptional simple Lie algebras, so called $E_{71/2}$ algebra and also two other similar (unknown) objects with positive dimensions. In addition, there are 47 isolated solutions in "unphysical semiplane" with negative dimensions. Isolated solutions mainly lie on a few straight lines in Vogel plane. All solutions give an integers in universal dimension formulae for first three symmetric powers of adjoint representation.