Let X be an irreducible smooth complex projective variety. Let G be a linear algebraic group over C. We define the notion of Lie algebroid valued connection on holomorphic principal G--bundles on X, and study their basic properties under extension and reduction of structure group. Finally we investigate criterions for existence of a Lie algebroid connection on principal G--bundles over smooth complex projective curves.
Ghosh et al. (Sat,) studied this question.