Description
The standard bosonic higher spin algebra in d dimensional anti-deSitter space AdS_d is given by the enveloping algebra of SO(d-1,2) quotiented by its Joseph ideal which defines its minimal unitary representation (minrep). The quasiconformal method provides us with a unified approach to minreps of non-compact groups and leads to realizations for which the Joseph ideal vanishes identically as operators. Hence the quasiconformal realization of the enveloping algebra of the minrep of SO(d-1,2) yields directly the bosonic higher spin algebra in AdS_d. There exists a one-to-one correspondence between the minrep of SO(d-1,2) and its deformations and massless conformal fields in d-dimensional Minkowski spacetimes. The enveloping algebras of the deformed minreps lead to novel deformed higher spin algebras in AdS_d. Furthermore, the quasiconformal approach allows one to define supersymmetric extensions of higher spin algebras and their deformations. I will review the quasiconformal approach to the unitary realizations and classification of higher spin algebras , their deformations and supersymmetric extensions in all dimensions.
