Moduli spaces of affine homogeneous spaces (Q2330475)

In the paper under review, the author studied formal affine homogeneous spaces defined as a pair \(\mathfrak{g} \supset \mathfrak{h}\) of Lie algebras endowed with a left-invariant connection \(A:\mathfrak{g} \rightarrow \operatorname{End} \mathfrak{g}/\mathfrak{h}\). In order to describe the moduli spaces of formal affine homogeneous spaces, one can fix a model vector space \(V\) and augment a given formal affine homogeneous space \(\mathfrak{g} \supset \mathfrak{h}\) endowed with a left-invariant connection \(A:\mathfrak{g} \rightarrow \operatorname{End} \mathfrak{g}/\mathfrak{h}\) by an isomorphism of frame \(F:V \rightarrow \mathfrak{g}/\mathfrak{h}\) and a split \(\mathfrak{g}/\mathfrak{h}\rightarrow \mathfrak{g}\) of the canonical projection. This construction gives the way to associate a connection-curvature-torsion triple \((A,R,T)\) on a vector space \(V\) to an augmented formal affine homogeneous space. Theorem 4.8 in the paper under review gives a criterion that a connection-curvature-torsion triple \((A,R,T)\) on a vector space \(V\) represent a formal affine homogeneous spaces. This result generalizes the well-known description of local isometry class of symmetric spaces in terms of their curvature \(R\) and the classification of manifolds with parallel curvature and torsion by \textit{W. Ambrose} and \textit{I. M. Singer} [Duke Math. J. 25, 647--669 (1958; Zbl 0134.17802)]. Using the above description of affine homogeneous spaces, the author constructed the moduli space \(\mathfrak{M}_\infty(\,\mathfrak{gl}\,V\,)\) for the local isometry classes of affine homogeneous spaces as the set of equivalence classes of points \((A,R,T)\in \mathfrak{M}(\,\mathfrak{gl}\,V\,)\) under the infinite-order contact relation \(\sim_{\infty}\). Infinitesimal deformations of an isometry class of affine homogeneous spaces in this moduli space are described by the Spencer cohomology of a comodule associated to a point in \(\mathfrak{M}_\infty(\,\mathfrak{gl}\,V\,)\).