The collineation groups of division ring planes. II. Jordan division rings

Robert Harvey Oehmke and Reuben Sandler

Vol. 15 (1965), No. 1, 259–265

DOI: 10.2140/pjm.1965.15.259

Abstract

In this paper the authors continue their study of the collineation groups of division ring planes (The collineation groups of division ring planes I. Jordan division algebras, J. Reine and Angew. Math. vol. 216, 1964). Some of the results obtained for finite dimensional Jordan division algebras are extended to a special class of infinite dimensional algebras.

As is well-known the study of the collineation group of a projective plane $\pi$ coordinatized by an algebra $\mathcal{ℛ}$ can be reduced to the stutiy of the autotopism group of $\mathcal{ℛ}$ or the group of autotopic collineations of $\pi$, $\mathcal{\mathscr{H}}\left(\pi \right)$. The pair $\left(a,b\right)$, $a,b\in \mathcal{ℛ}$, is defined to be admissible if and only if there exists an element $\alpha$ in $\mathcal{\mathscr{H}}\left(\pi \right)$ with $\left(1,1\right)\alpha =\left(a,b\right)$. Modulo the automorphism group of $\mathcal{ℛ}$, the determination of $\mathcal{\mathscr{H}}\left(\pi \right)$ is equivalent to the determination of all admissible pairs $\left(a,b\right)$ and coset representatives ${\phi }_{a,b}\in \mathcal{\mathscr{H}}\left(\pi \right)$ such that $\left(1,1\right){\phi }_{a,b}=\left(a,b\right)$. With either the assumption $\mathcal{ℛ}$ algebraic over its center, or the assumptions characteristic of $\mathcal{ℛ}$ not equal to 0 and the centers of $\mathcal{ℛ}$ and ${\mathcal{ℛ}}^{\prime }$ (the algebra of all elements of $\mathcal{ℛ}$ algebraic over the center of $\mathcal{ℛ}$) equal, the admissible pairs $\left(a,b\right)$ are determined. Use is made of Kleinfeld’s result on the middle nucleus of Jordan rings (Middle nucleus = center in a simple Jordan ring, to appear.) We also prove and use the result that the algebra $\mathcal{ℰ}$ consisting of all right multiplications $R_f$ is commutative, where $f$ is in the subalgebra generated by $a$ and $a^{-1}$ over the base field.

Mathematical Subject Classification

Primary: 17.40

Secondary: 50.70

Milestones

Received: 23 January 1964

Published: 1 March 1965