Classification of quaternionic spaces with a~transitive solvable group of motions

D. V. Alekseevskii

Izvestiya. Mathematics , Tome 9 (1975) no. 2, pp. 297-339

Résumé

A complete classification of quaternionic Riemannian spaces (that is, spaces $\mathscr V^n$ with the holonomy group $\Gamma\subset Sp(1)\cdot Sp(m)$, $n=4m$), which admit a transitive solvable group of motions is given. It turns out that the rank of these spaces does not exceed four and that all spaces $\mathscr V^n$ whose rank is less than four are symmetric. The spaces $\mathscr V^n$ of rank four are in natural one-to-one correspondence with the Clifford modules of Atiyah, Bott and Shapiro. In this correspondence, the simplest Clifford modules, which are connected with division algebras, are mapped to symmetric spaces of exceptional Lie groups. Other Clifford modules, which are obtained from the simplest with help of tensor products, direct sums and restrictions, correspond to nonsymmetric spaces. Bibliography: 17 items.