John Browne

Grassmann Algebra John Browne

This book is an abridged draft edition. It is being published now in this form due to circumstances beyond the author's control. There will not be another edition. You will miss some things you might have expected: page numbers in the table of contents, a comprehensive bibliography and references, an index, a well-reviewed text .. . This book (Volume 2) follows on from Volume 1 so you may also need Volume 1 for reference. The book shows how the hypercomplex and associative algebras are hidden in Grassmann algebra. It shows how the quaternions and octonions and their split variants, and the geometric and Clifford algebras, are simply Grassmann algebra. It does this by extending its two familiar product operations, the exterior and interior products to define a suite of products together called the generalized Grassmann product. It shows how hypercomplex, geometric and Clifford products may then be defined as linear combinations of generalized Grassmann products in which the scalar coefficients are restricted to unity or negative unity only. This binary variability is sufficient to endow properties to a product operation, for example associativity. The book concludes by finding four associative product operations, two of which being the geometric and Clifford products. But these work on Grassmann entities only, so they are operations of the Grassmann algebra. In sum: this book shows how some important linear algebras such as the hypercomplex, geometric and Clifford algebras can be constructed entirely within Grassmann algebra by defining specialized product operations using only the exterior and interior products.