Grassmann–Cayley algebra

{{see also|Geometric algebra}}

In mathematics, a Grassmann–Cayley algebra is the exterior algebra with an additional product, which may be called the shuffle product or the regressive product.{{refn|

{{citation

| last = Perwass | first = Christian

| isbn = 978-3-540-89067-6

| mr = 2723749

| page = 115

| publisher = Springer-Verlag, Berlin

| series = Geometry and Computing

| title = Geometric algebra with applications in engineering

| url = https://books.google.com/books?id=8IOypFqEkPMC&pg=PA115

| volume = 4

| year = 2009| bibcode = 2009gaae.book.....P

}}}}

It is the most general structure in which projective properties are expressed in a coordinate-free way.{{refn|

{{citation

| author1 = Hongbo Li

| first2 = Peter J. |last2=Olver

| year = 2004 |isbn=9783540262961

| title = Computer Algebra and Geometric Algebra with Applications: 6th International Workshop, IWMM 2004, GIAE 2004

| publisher = Springer |volume=3519 |series=Lecture Notes in Computer Science

| url = https://books.google.com/books?id=q68fUw31mrkC&pg=PA387

}}}}

The technique is based on work by German mathematician Hermann Grassmann on exterior algebra, and subsequently by British mathematician Arthur Cayley's work on matrices and linear algebra.

It is a form of modeling algebra for use in projective geometry.{{cn|date=September 2017}}

The technique uses subspaces as basic elements of computation, a formalism which allows the translation of synthetic geometric statements into invariant algebraic statements. This can create a useful framework for the modeling of conics and quadrics among other forms, and in tensor mathematics. It also has a number of applications in robotics, particularly for the kinematical analysis of manipulators.