The role of nonassociative algebra in projective geometry (Q2927911)

This book presents a very focused derivation of the fascinating connections between the algebraic concepts field, associative division ring, and alternative division ring on the one hand and the geometric concepts Pappus plane, dilatation plane, and transvection plane on the other hand; all this is contained in the first culmination point of the book, namely in Theorem 5.18, where the reader finds a well-arranged survey done in exemplary fashion. The chosen way to the mentioned theorem is short, elegant, and seems to be new. To give a rigorous introduction of the notion ``dimension'' the author prefers the lattice-theoretical approach. The thorough and detailed investigation of the exceptional octonion plane is prepared by chapters on: NEWLINENEWLINE\quad(1) automorphisms of a projective geometry \({\mathcal G}(V)\) for a vector space \(V\) over an associative division ring \(\Delta\) NEWLINENEWLINE\quad(2) quadratic forms and orthogonal groups NEWLINENEWLINE\quad(3) homogeneous maps NEWLINENEWLINE\quad(4) norms and Hermitian matrices.NEWLINENEWLINEThroughout the book the author includes, wherever possible, the cases with characteristic \(2\) and those with infinite dimension. In the penultimate chapter an axiomatic study of incidence planes with remoteness relation is undertaken. The last chapter outlines some connections of nonassociative algebra to other geometries, including buildings.NEWLINENEWLINEThe text addresses university teachers and graduate students who have a knowledge of linear algebra, basic ring theory, and basic group theory. The reader benefits from the author's long experience in research and lecturing of algebra and geometry. With this book the author takes the student towards research areas. The proofs, except those of the sketchy Chapter 14, are complete and done with contemporary mathematical rigor. Many interesting topics, that would have hindered the progress, are contained in the Exercises.NEWLINENEWLINEThe text is divided into 14 chapters, each begins with a preview and ends with exercises.NEWLINENEWLINEChapter 1 (Affine and projective planes) deals with incidence geometry, homomorphisms, isomorphisms, axioms defining affine planes, axioms defining projective planes, quadrangles, diagonal points, projections of \(\ell\) to \(\ell'\) from \(P\), duality, dual homomorphisms, and polarity. This chapter provides exercises on vector spaces over an associative division ring, the order of a finite plane, and the Fano condition.NEWLINENEWLINEIn Chapter 2 (Central automorphisms of projective planes), center, axis, transvection (= elation), dilatation (= homology), transitivity properties, the transvection plane, and the dilatation plane are considered. Topics of the exercises are reflection and the reflection plane.NEWLINENEWLINEChapter 3 (Coordinates for projective planes) considers ternary systems, standard coordinatization, associated sum, associated product, additive identity, nullity, the Cartesian group \(\mathcal C\), two coordinatizations related to the projective plane \({\mathcal G}({\mathcal C})\), transvections and algebraic properties, left (right) distributive, Veblen-Wedderburn systems, unital rings, division rings, special Jordan rings, left (right and middle) Moufang condition, Moufang rings, alternative rings, associator, (table on p. 39:) juxtaposition of geometric properties and corresponding algebraic properties. The exercises deal with central duality and the central duality plane.NEWLINENEWLINEChapter 4 (Alternative rings) presents left Moufang rings, Jordan polynomials, commutators, shows that a left Moufang division ring is alternative, consideres Artin's theorem, left (middle, right) nucleus, nucleus, the center of a ring, inverses in alternative rings, the Cayley-Dickson process, quadratic extensions, composition algebras, nondegenerate quadratic extensions, quaternion algebras, octonion algebras, and split composition algebras. It presents the proof (Skornyakov or Bruck and Kleinfeld:) that an alternative division ring is either associative or an octonion algebra over its center. There are exercises on the Zorn matrix algebra and Wedderburn's little theorem.NEWLINENEWLINEThe topics of Chapter 5 (Configuration conditions) are the Desargues condition, the little Desargues condition, the pre-Desargues configuration, quadrangle sections, quadrangle section conditions, little quadrangle section conditions, reversed quadrangle section conditions, Pappus' condition, the fact that the Pappus condition implies the Desargues condition, configurations and central automorphisms, paths of length \(n\), loops, homotopy sets, left cancellation, rotation, simply connectedness, and Theorem 5.18. (see above). There are exercises on harmonic condition.NEWLINENEWLINEChapter 6 (Dimension theory) introduces the closure family, span, irreducibility, the direct span, finitely closed families, dimensionable families, independence and bases, and strongly dimensionable sets. Exercises on dependance relations and the Steinitz exchange axiom are given.NEWLINENEWLINEIn Chapter 7 (Projective geometries), axioms of a projective geometry, axioms of a nearly projective geometry, the relation to strongly dimensionable sets, the classification of projective geometries, planes, hyperplanes, central automorphisms, centers, axes, transvections (= elations), dilatations (= homology), and central automorphism geometry are considered. If the projective dimension of a projective geometry \(\mathcal G\) is \(\geq\,3\), then \({\mathcal G}\cong{\mathcal G}(V)\) for some vector space \(V\) over an associative division ring.NEWLINENEWLINEChapter 8 (Automorphisms of \({\mathcal G}(V)\)) deals with the fundamental theorem of projective geometry, dual spaces, properties of the projective elementary group \(\mathrm{PE}(V)\), simple groups, primitive groups, stabilizer subgroups, and derived groups. If \(V\) is a vector space over an associative division ring \(\Delta\) and \(\dim V\geq 3\), then \(\mathrm{PE}(V)\) is a simple group.NEWLINENEWLINEChapter 9 (Quadratic forms and orthogonal groups) is about nondegenerate forms, isotropic subspaces, hyperbolic pairs, hyperbolic spaces, dual bases, the Witt index, orthogonal groups of a quadratic form on vector space, reflections, Eichler transformations, and the Cartan-Dieudonné theorem. This chapter provides exercises on the topic special orthogonal groups.NEWLINENEWLINEChapter 10 (Homogeneous maps) introduces (in a basis-free approach) polarization of a homogeneous map, homogeneous maps \(f\) of degree \((n_1,\dots,n_r)\), and semisimilarity of \(f\) with multiplier.NEWLINENEWLINEIn Chapter 11 (Norms and Hermitian matrices), the authors look at a version of the determinant, which we call the norm, for Hermitian matrices over a composition algebra \(\mathcal C\). \(\mathcal C\) is assumed to be a composition algebra over a field \(K\), \({\mathcal H}({\mathcal C}_n)\) is the set of Hermitian matrices with diagonal entries in \(K\). Further topics are the Hermitian elementary group \(\mathrm{HE}_n({\mathcal C})\), norms on \({\mathcal H}({\mathcal C}_n)\), transitivity of \(\mathrm{HE}_n({\mathcal C})\), trace and adjoint, and properties of \(\mathrm{HE}_3({\mathcal O})\) where \(\mathcal O\) is an octonion algebra. Exercises on Pfaffians are given.NEWLINENEWLINEChapter 12 (Octonion planes) gives an overview on constructions of octonion planes, Hermitian matrix planes, remoteness relations, the projective Hermitian elementary group \(\mathrm{PHE}_3({\mathcal O})\), the simplicity of \(\mathrm{PHE}_3({\mathcal O})\), and automorphisms of octonion planes.NEWLINENEWLINEChapter 13 (Projective remoteness planes) presents definition and examples. If \(\mathcal A\) is a unital associative ring, then \({\mathcal G}({\mathcal A}^3)\) is a projective remoteness plane if and only if \(\mathcal A\) is a two-sided inverse ring. Groups of Steinberg type, a criterion that a group of Steinberg type is parametrized by a unital nonassociative ring, the definition of a Moufang element of a ring, and transvections in a connected projective remoteness plane are also considered. Exercises deal, e.g., with the Barbilian plane.NEWLINENEWLINEIn Chapter 14 (Other geometries), the authors introduce the concepts necessary to define a spherical building in the language of abstract simplicial complexes, the Erlangen program, and the geometry of \(R\)-spaces. Because the building approach has many advantages, the geometry of \(R\)-spaces is more intuitive, they use it as transition to buildings. Further topics are the projective case, the orthogonal case, the symplectic case, buildings, abstract simplicial complexes, Coxeter complexes, spherical buildings, apartments, generalized \(n\)-gons, the Moufang-\(n\)-gon, Moufang sets and structurable algebras, linear Jordan algebras, Freudenthal-Tits magic squares, elliptic planes (and elliptic polarity) over the reals, the complex numbers, the quaternions and the octonions, respectively, and rotational symmetric space.