Abstract:

The recent achievements in the scope of Vogel’s universal approach to simple Lie algebras, which constitute the contents of my P.h.D. thesis will be presented. Particularly, I will

• present a new series of universal quantum dimensions of the simple Lie algebras, namely for the Cartan products of arbitrary powers of the adjoint (ad) and the representations, as well as a universal expression for the eigenvalues of the second Casimir operator on them.
• prove a feature, called “linear resolvability” of all known universal quantum dimension formulae, which relates to its possible singularities.
• formulate the problem of uniqueness of universal formulae, which connects them with so-called geometrical configurations of points and lines, and present explicit non-uniqueness factors.
• present a generalization of the Kac-Peterson formula identity for the determinant of the symmetrized Cartan matrix, leading to the explicit presentation of the partition function of the refined Chern-Simons theory on S³ for an arbitrary gauge group. This will make it possible to establish CS/topological strings dualities for all simple groups.

(based on the PhD thesis)