Conformal and quasiconformal realizations of exceptional Lie groups

We present a nonlinear realization of E₈₍₈₎ on a space of 57 dimensions, which is quasiconformal in the sense that it leaves invariant a suitably defined "light cone" in ℝ⁵⁷. This realization, which is related to the Freudenthal triple system associated with the unique exceptional Jordan algebra over the split octonions, contains previous conformal realizations of the lower rank exceptional Lie groups on generalized space times, and in particular a conformal realization of E₇₍₇₎ on ℝ²⁷ which we exhibit explicitly. Possible applications of our results to supergravity and M-Theory are briefly mentioned.