Norbert Knarr

Polarities of shift planes

We construct polarities for arbitrary shift planes and develop criteria for conjugacy under the normalizer of the shift group. Under suitable assumptions (in particular, for finite or compact planes) we construct all shift groups on a given plane, and our constructions yield all conjugacy classes of polarities. We show that a translation plane admits an orthogonal polarity if, and only if, it is a shift plane. The corresponding planes are exactly those that can be coordinatized by commutative semifields. The orthogonal polarities form a single conjugacy class. Finally, we construct examples of compact connected shift planes with more conjugacy classes of polarities than the corresponding classical planes.

Polarities and unitals in the Coulter---Matthews planes

For a class of finite shift planes introduced by Coulter and Matthews, we give a set of representatives for the isomorphism types, determine all automorphisms and describe all polarities explicitly. The planes in question are the only known examples of finite shift planes that are not translation planes. Each non-desarguesian Coulter–Matthews plane admits precisely two conjugacy classes of orthogonal polarities. In addition, each Coulter–Matthews plane of square order admits exactly one conjugacy class of unitary polarities. We prove that most of the corresponding unitals are not classical.

Unitals over Composition Algebras

Abstract. We consider polar unitals in projective planes over composition algebras. We prove that the little projective group of such a unital is contained in the centralizer of the polarity defining the unital. This yields information about the full automorphism group of the unital; it is contained in the automorphism group of the normal subgroup generated by all translations of the unital. Over suitable ground fields (such as the real numbers) we obtain a complete description of the automorphism group of the unital in the sense that it coincides with the centralizer of the polarity. The action of a nilpotent regular normal subgroup is used to establish isomorphisms between unitals in spaces of different dimensions, and over different fields.

Quasifields of symplectic translation planes

We show that a translation plane is symplectic if and only at least one of its associated quasifields admits a non-degenerate invariant symmetric bilinear form. As an application we prove that a proper desarguesian, Moufang or nearfield plane can never be symplectic. Moreover, we give a purely algebraic characterization of the quasifields which coordinatize symplectic translation planes.

4-Dimensionale projektive Ebenen mit grosser abelscher Kollineationsgruppe

We show that a 4-dimensional connected abelian group can act in exactly five different ways as a collineation group of a compact 4-dimensional projective plane. Furthermore the complex projective plane is characterized as the only compact 4-dimensional projective plane which admits two different 4-dimensional abelian collineation groups.

Polar spaces, BLT-sets and generalized quadrangles

Abstract We generalize an earlier construction of generalized quadrangles from polar spaces of rank 3 to polar spaces of arbitrary rank.

Transitive groups of similitudes generated by octonions

Abstract We study the structure of various groups generated by (left) multiplications in alternative division algebras, and construct epimorphisms onto orthogonal and special orthogonal groups. Throughout, we include the characteristic two case in our treatment.