Differential Invariants of the 2D Conformal Lie Algebra Action

We consider the Lie algebra that corresponds to the Lie pseudogroup of all conformal transformations on the plane. This conformal Lie algebra is canonically represented as the Lie algebra of holomorphic vector fields in ℝ²≃ℂ. We describe all representations of mathfrakgmathfrak{g} via vector fields in J ⁰ℝ²=ℝ³(x,y,u), which project to the canonical representation, and find their algebra of scalar differential invariants.