GENERALIZED CONFORMAL AND SUPERCONFORMAL-GROUP ACTIONS AND JORDAN ALGEBRAS

We study the ''conformal groups'' of Jordan algebras along the lines suggested by Kantor. They provide a natural generalization of the concept of conformal transformations that leave two-angle invariant to spaces where ''p-angle'' (p greater-than-or-equal-to 2) can be defined. We give an oscillator realization of the generalized conformal groups of Jordan algebras and Jordan triple systems. A complete list of the generalized conformal algebras of simple Jordan algebras and Hermitian Jordan triple systems is given. These results are then extended to Jordan superalgebras and super Jordan triple systems. By going to a coordinate representation of the (super)oscillators one then obtains the differential operators representing the action of these generalized (super) conformal groups on the corresponding (super) spaces. The superconformal algebras of the Jordan superalgebras in Kac's classification is also presented.