Minimal representations and reductive dual pairs in conformal field theory

A minimal representation of a simple non-compact Lie group is obtained by “quantizing” the minimal nilpotent coadjoint orbit of its Lie algebra. It provides context for Roger Howe’s notion of a reductive dual pair encountered recently in the description of global gauge symmetry of a (4-dimensional) conformal observable algebra. We give a pedagogical introduction to these notions and point out that physicists have been using both minimal representations and dual pairs without naming them and hence stand a chance to understand their theory and to profit from it. Invited lecture at the 8th International Workshop on Lie Theory and Its Applications in Physics, Varna, June 2009; ed. V. Dobrev, AIP Conference Proceedings 1243 (Melville, NY 2010) pp. 13-30. Permanent address: Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Tsarigradsko Chaussee 72, BG-1784 Sofia, Bulgaria; e-mail: [email protected]