Advanced Search

We shall show that no reductive splitting of the spin group exists in dimension $3\le m\le 20$ other than in dimension $m=4$. In dimension $4$ there are reductive splittings in any signature. Euclidean and Lorentzian signatures are reviewed in particular and signature $(2,2)$ is investigated explicitly in detail. Reductive splittings allow to define a global $\text{SU}\left(2\right)$-connection over spacetime which encodes in an weird way the holonomy of the standard spin connection. The standard Barbero-Immirzi (BI) connection used in LQG is...