On higher spin realizations of K(E10)

Abstract

Starting from the known unfaithful spinorial representations of the compact subalgebra K(E10) of the split real hyperbolic Kac-Moody algebra E10 we construct new fermionic `higher spin' representations of this algebra (for `spin-5/2' and `spin-7/2', respectively) in a second quantized framework. Our construction is based on a simplified realization of K(E10) on the Dirac and the vector spinor representations in terms of the associated roots, and on a re-definition of the vector spinor first introduced by Damour and Hillmann. The latter replaces manifestly SO(10) covariant expressions by new expressions that are covariant w.r.t. SO(1,9), the invariance group of the DeWitt metric restricted to the space of scale factors. We present explicit expressions for all K(E10) elements that are associated to real roots of the hyperbolic algebra (of which there are infinitely many), as well as novel explicit realizations of the generators associated to imaginary roots and their multiplicities. We also discuss the resulting realizations of the Weyl group.