On dimensions of affine transformation groups transitively acting

A. V. Loboda

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 4 (2014), pp. 11-35.

Résumé

The main content of the article constitute the three theorems related to the dimensions of homogeneous manifolds. Traditionally we mean homogeneity as the existence of a local Lie group, that acts transitively on the manifold under consideration near a selected point. As acting groups, in affine homogeneity case only a subgroups of the group Aff $ (3, \Bbb C)$ are considered. The main instruments of the article are the affine canonical equations for the studied homogeneous surfaces and the Lie algebras of affine vector fields tangent to these manifolds. The concept of affine homogeneity is closely related, in the case of real hypersurfaces, to holomorphic homogeneity, that is natural for multidimensional complex analysis. But even for the hypersurfaces of $3$-dimensional complex spaces the classification problems in both cases (affine and holomorphic) do not arise until the complete solution. One of the things that could help to obtain such a solution is to understand the situation with possible dimensions of a Lie groups (and Lie algebras) which acts transitively on the homogeneous manifolds under consideration. In this paper the dependence is studied of such dimension from a couple of the Taylor coefficients of the $2$-nd order (specifying the type of surface) of the canonical equation of strictly pseudo convex (SPC) hypersurface. In this article we obtain (in Theorem 1) a general estimate for the dimension of such groups for an arbitrary strictly pseudo convex affinely homogeneous hypersurfaces in complex space $ \Bbb C^3 $. For one of the several types of homogeneous surfaces, this estimate is sharp: the affine transformation group with the maximal possible dimension $10$ acts transitively on the quadric $ Im \, w = | z_1 |^ 2 + | z_2 |^ 2 $. For affinely homogeneous surfaces, that are not equivalent to this quadric, the dimension of such affine group does not exceed $7$. The proof of this statement is given in the article (in Theorem 2) for the surfaces of the type $(1/2, 0)$. To prove this assertion we use the coefficients structure of affine vector fields tangent to homogeneous surfaces. The description of this structure is also obtained in the paper. In Theorem 3, the describing problem for the affine homogeneous surfaces of the type $(1/2,0)$ with «rich» symmetry groups in space $ \Bbb C^3 $ is reduced to the study of only $5$-dimensional Lie groups and algebras. Note that in the general context of the homogeneity problem such a reduction is not possible; here the specificity of the studied type of the surfaces plays the crucial role. Note that a complete description of affinely homogeneous hypersurfaces of the type $(1/2, 0)$ in the space $ \Bbb C^3 $ with $5$-dimensional algebras of tangent vector fields is presented in a joint paper of the author (ArXiv.org, 2014). Because of the sufficiently large dimensions the scheme of the homogeneous varieties description, as well as this article considerations, imply significant use of computer symbolic calculations.

Détail

Mots-clés : affine transformation
Keywords: homogeneous manifold, vector field, Lie algebra, canonical equation of surface.

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