Universal and homogeneous embeddings of dual polar spaces of rank 3 defined over quadratic alternative division algebras (Q294259)

For every alternative division algebra \(O\), which is quadratic over a central subfield \(K\), there is associated a projective plane and its dual and a polar space and its dual.NEWLINENEWLINEIn particular, the polar space is described in great detail. With \(n\) the degree of \(O\) over \(K\), the authors give an explicit representation of the dual polar space into the projective space of dimension \((6n+7)\) over \(K\). If \(| K| >2\), then this embedding is universal. The case \(| K| =2\) is also treated in detail.NEWLINENEWLINEThe embedding induces a representation of the little projective group of the polar space (or equivalently its dual) into the \(\mathrm{PGL}(6n+8, K)\). Generators for this representation are given.