Geometry and Mathematical Physics Seminar 2011/2012

Universal, in the sense of Vogel's Universal Lie Algebra, expression for the generating function of the spectra of Casimir's operators on adjoint representation is presented, for two choices of Casimir operators. Generalized universal Freudenthal-de Vries strange relations are derived. They lead to a universal representation of (the perturbative part of) the $S_{00}$ element of the modular transformations matrix (= the partition function of the Chern-Simons theory on a $3$-sphere) of characters of Kac-Moody algebra. Extension of universality to the untwisted Kac-Moody algebras is discussed and the "extended Vogel space" $P^3/S_3$ is introduced. The $n\to -n$ duality of the Lie algebras $so(2n)$ and $sp(2n)$ is extended to the classical symmetric spaces using the Macdonald duality for the Jack and Jacobi symmetric functions. The explicit formulae for the Popov-Perelomov generating function of Casimir's spectra are shown to satisfy the duality.

The seminar will consist of two independent parts:

• 3pm-3.50pm: introductory lecture accessible for students;
• 4.10pm-5.10pm: main talk (independent of the introductory part).