 Invariant frames of the real affine group

ID: invariant-frames-of-the-real-affine-group

In exponential scale coordinates, the real affine group has matrices

(e^{α} 0 β 1)

. The left and right Maurer-Cartan forms are respectively

D d α + T e^{- α} d β

and

D d α + T (d β - β d α)

. Their dual vector fields obey

[D_{L}, T_{L}] = T_{L}

and

[D_{R}, T_{R}] = - T_{R}

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