Canonical connections

We study the space of canonical connections on a reductive homogeneous space. Through the investigation of lines in the space of connections invariant under parallelism, we prove that on a compact simple Lie group, bi-invariant canonical connections are exactly the bi-invariant connections that are invariant under parallelism. This motivates our definition of a family of canonical connections on Lie groups that generalizes the classical $(+)$, $(0)$, and $(-)$ connections studied by Cartan and Schouten. We find the horizontal lift equation of each connection in this family, as well as compute the square of the corresponding Dirac operator as the element of non-commutative Weil algebra defined by Alekseev and Meinrenken.\u