Lifting skewfields

Abstract

There is a well known algebraic process by which spreads and quasifibrations with their representative partial spreads are within PG(3, K), where K is a field that admits a quadratic field extension may be “lifted” to a corresponding spread or quasifibration, respectively. The lifted quasifibrations are derivable maximal partial spreads within $PG(3,K(\tau ))$, where $K(\tau )$, denotes the associated quadratic field extension of K. In this article, we ask whether there is any sort of “lifting” process for spreads written over non-commutative skewfields. We consider where the lifting theory might extend to quasifibrations that live in PG(3, K), where K is considered to be a non-commutative skewfield. We are able to show that any non-commutative skewfield that admits a central Galois quadratic extension skewfield always may be lifted to a new semifield. This occurs, in particular, in most quaternion skewfield planes and since there is a matrix representation of a quaternion skewfield within GL(3, K), for certain fields K, there are then two mutually non-isomorphic skewfield planes constructed in mutually non-isomorphic 3-dimensional projective spaces. Taken purely algebraically, this lifting construction is similar to what might be called a Cayley–Dickson procedure, as the constructed semifields, although not algebras, are the resultant when applying this procedure to a central Galois quadratic extension quaternion division ring, thereby losing associativity.