Classical groups and geometric algebra (Q2756812)

The classical groups in question are the general linear, symplectic, orthogonal and unitary groups over an arbitrary field. The aim of the book is to acquaint the reader with their basic algebraic and geometrical properties in a direct, simple and uniform way. Designed as a textbook for a graduate-level course, it is attratively written and clearly set out. The theory proper occupies 15 chapters while some further developments are rapidly sketched in a final chapter.NEWLINENEWLINENEWLINEA typical classical group \(G\) has a finite normal series \(G=G_0\supset G_1\supset\cdots\supset G_k=\langle 1\rangle\) in which one quotient group \(G_i/G_{i+1}\) is non-Abelian simple and the rest are Abelian. Briefly put, the mathematical goal of the book is to determine this series and its quotients. Geometric algebra, to which the greater part of the book is devoted, provides the means of doing this. In the tradition of \textit{E. Artin}'s ``Geometric Algebra'' (1957; Zbl 0077.02101) this subject is developed ab initio using only basic linear algebra and group theory. The author's unified treatment of the different kinds of classical group depends on a simple but effective theorem of Iwasawa about primitive permutation groups. In this context, the role of geometric algebra is to show that the conditions of Iwasawa's Theorem are satisfied.NEWLINENEWLINENEWLINESeveral points may be noted. The theory of the orthogonal groups over fields of characteristic 2 differs markedly from the theory over fields of characteristic \(\neq 2\). For clarity of exposition, the author devotes 3 separate chapters to the former. As is well known, the uniform treatment of the orthogonal and unitary groups has to be restricted to the case where the Witt index of the underlying quadratic or Hermitian form is positive. Since the orthogonal groups of Euclidean space escape the net, the author devotes a special chapter to them. Finally, 2 chapters are allotted to the Clifford algebras, partly for their intrinsic interest and partly because of their connections with spin representations and the spinor norm.NEWLINENEWLINENEWLINEAs indicated above, the author's approach in the present book is direct and essentially elementary. Such more advanced or technical topics as Lie theory, Chevalley groups, BN-pairs and buildings are briefly mentioned but form no integral part of the text. What is done in the book is done very well and very thoroughly. The reader who masters it will have a sound basic knowledge at the classical groups.