Quasifields of symplectic translation planes
Let \(V\) be a vector space over the field \(F\) and let \(\Sigma\) be a spread on \(V\), i.e.\ a covering of \(V\) by a collection of mutually complementary subspaces. If every element of \(\Sigma\) is totally isotropic w.r.t a non-degenerate symplectic bilinear form \(S : V\times V \rightarrow F\), then \(\Sigma\) is called a symplectic spread. Starting with a quasifield \(Q\) and a subfield \(F\) of the kernel \(K\) of \(Q\), the author shows that symplectic forms \(Q^2\times Q^2\rightarrow F\) turning \(\Sigma\) into a symplectic spread are in 1-1 correspondence to non-degenerate invariant bilinear forms \(B:Q\times Q\rightarrow F\) (i.e.\ \(B(xa,y)=B(x,ya)\) holds for all \(a,x,y\in Q\)) which in turn are in 1-1 correspondence to certain linear maps \(Q\rightarrow F\). If \(Q\) admits a non-degenerate invariant form, then the kernel \(K\) is commutative. Moreover, the right and the middle nucleus of \(Q\) coincide and both are contained in the center. From these results it follows that if a Desarguesian, Moufang or nearfield plane is symplectic, then the plane is actually Pappian.
scientific article; zbMATH DE number 5566223