On holomorphic realizations of 5-dimensional Lie algebras

Realizations are studied for a particular block of 5-dimensional Lie algebras (within the well-known Mubarakzyanov classification) in the form of algebras of holomorphic vector fields on homogeneous real hypersurfaces of the 3-dimensional complex space. All (locally) holomorphically homogeneous and Levi nondegenerate real hypersurfaces associated with algebras in the block in question are described. A majority of such manifolds are holomorphic images of tubular hypersurfaces with affine homogeneous base. At the same time, two new holomorphically homogeneous hypersurfaces are obtained that do not reduce to tubes, have sign-indefinite Levi form, and are algebraic surfaces of degree 3.