The geometry and algebra of the representations of the Lorentz group

In this paper a (2j + 1)-spinor analysis is developed along the lines of the 2-spinor and 3-spinor ones. We define generalized connecting quantities A$^{\mu\nu}$(j) which transform like (j,0) $\bigodot$ (j-1, 0) in spinor space and like second rank tensors under transformations in space-time. The general properties of the A$^{\mu\nu}$ are investigated together with algebraic relations involving the Lorentz group generators, J$^{\mu\nu}$. The connexion with 3j symbols is discussed. From a purely formal point of view we introduce a geometrical representation of a (2j + 1)-spinor as a point in a 2j dimensional projective space. Then, for example, the charge conjugate of a (2j + 1)-spinor is just the polar of the corresponding point with respect to a certain rational, normal curve in the projective space. It is suggested that this representation will prove useful.