On a Family of Lie Algebras Related to Homogeneous Surfaces

Real affine homogeneous hypersurfaces of general position in three-dimensional complex space $\mathbb C^3$ are studied. The general position is defined in terms of the Taylor coefficients of the surface equation and implies, first of all, that the isotropy groups of the homogeneous manifolds under consideration are discrete. It is this case that has remained unstudied after the author's works on the holomorphic (in particular, affine) homogeneity of real hypersurfaces in three-dimensional complex manifolds. The actions of affine subgroups $G\subset \mathrm {Aff}(3,\mathbb C)$ in the complex tangent space $T_p^{\mathbb C}M$ of a homogeneous surface are considered. The situation with homogeneity can be described in terms of the dimensions of the corresponding Lie algebras. The main result of the paper eliminates “almost trivial” actions of the groups $G$ on the spaces $T_p^{\mathbb C}M$ for affine homogeneous strictly pseudoconvex surfaces of general position in $\mathbb C^3$ that are different from quadrics.