I am reading a physics paper, in which the Lie derivative is presented in this strange (to me) way: $$\mathcal{L}_Y=Y^A\partial_A+\frac{i}{2}D_AY_BS^{AB}$$ where $A,B=\{z,\bar{z}\}$ with $\{z,\bar{z}\}$ being the coordinates on the complex sphere (i.e. connected to the usual polar/azimuthal coordinates via $z=e^{i\phi}\tan\frac{\theta}{2}$) and $S_{AB}$ is the pullback of the spin operator on the complex two sphere. The antisymmetric spin operator is given by $$S_{12}=\frac{h(1-z\bar{z})}{1+z\bar{z}},\ S_{13}=\frac{ih(z-\bar{z})}{1+z\bar{z}},\ S_{23}=\frac{h(z+\bar{z})}{1+z\bar{z}}$$ where the indices run from zero to three. Can someone shed some light on how do we manage to write the Lie derivative without the tensor it is supposed to be acted upon and on how can one write it in terms of the pullback of some operator?? I do not know if my question makes sense, but basically I can not understand why the Lie derivative is given in the form it is given.

I have taken a geometry of GR class, so I am familiar with the basic definitions and with finding the components of the Lie derivative, when acted upon an arbitrary tensor. Thanks a lot.



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