Abstract
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.
Highlights
A minimal representation of a simple non-compact Lie group is obtained by “quantizing” the minimal nilpotent coadjoint orbit of its Lie algebra
One can ask how does an irreducible representation (IR) of G split into IRs of the pair (G1, G2)
We introduce there the notion of a bilocal field which opens the way to use infinite dimensional Lie algebras in the context of 4D CFT
Summary
“. . . for me the motivation is mostly the desire to understand the hidden machinary in a striking concrete example, around which one can build formalisms”.3. It was previewed in a lecture [H85] highlightening its applications to physics The principle underlying such applications was, anticipated by the founders of the axiomatic approach to quantum field theory who introduced the concept of superselection sectors [W3] and developed it systematically in the work of Doplicher, Haag and Roberts (for a review and further references – see [H]). In this framework the algebra of observables and the global (compact) gauge group are mutual commutants in the (extended) field algebra, providing a generic example of a dual pair.
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