Abstract
We consider formulae for universal (in Vogel’s sense) quantum dimensions and their applications. Universal quantum dimensions of Cartan powers of $${{X}_{2}}$$ and adjoint representations at the points of special type singularities are shown to give correct answers when restricted to both exceptional and one other appropriate lines. Derivation of Diophantine equations, classifying simple Lie algebras, from the universal quantum dimension of the adjoint representation is presented.
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