Grassmann Algebra starts with a vector space (or more generally a module) of dimension 'n' and from it generates a vector space 'A' of dimension 2n or, another way to think about it, the vector space 'A' is made up of a number of smaller dimensional vector spaces.
The elements in Grassman Algebra consists of a compound element made up from parts of different 'grades':
| Grade | Description |
|---|---|
| 0 | Real Numbers (or whatever the base element over which the vectors are based) |
| 1 | Vectors (or more generally modules) with a given dimension 'n'. |
| >1 | Higher order bases 'generated' from the above bases by the multiplication rules which we will describe. |
Addition in Grassmann algebras is always defined by adding corresponding terms, this is relatively simple in that, it does not involve interaction between the various components.
So in order to define a Grassmann Algebra we must define multiplication rules. We require the algebra operations to be 'closed' and finite, that is, when two elements of the algebra are combined the result is another element of the algebra. For example a complex number multiplied by another complex number will always produce another complex number. Grassmann and Clifford Algebra has a number of sub-algebras within one big algebra, however the sub-algebras are not closed within themselves only the overall algebra is closed. Apart from the case where grade=0 which is just the real numbers all clifford algebras have a subset which are vectors, but multiplying a vector by another vector produces something that is not a vector, we can think of this as generating other types.
The number and dimension of the vector spaces have the structure of the binomial triangle. For any dimension 'n' of the vector space that we start with the vector spaces are given by the following sequence:
| n | 2n= | ||||||||||||||||||
| 0 | 1 | 1 | |||||||||||||||||
| 1 | 1 | 1 | 1+1 | ||||||||||||||||
| 2 | 1 | 2 | 1 | 1+2+1 | |||||||||||||||
| 3 | 1 | 3 | 3 | 1 | 1+3+3+1 | ||||||||||||||
| 4 | 1 | 4 | 6 | 4 | 1 | 1+4+6+4+1 | |||||||||||||
| 5 | 1 | 5 | 10 | 10 | 5 | 1 | 1+5+10+10+5+1 | ||||||||||||
| 6 | 1 | 6 | 15 | 20 | 15 | 6 | 1 | 1+6+15+20+15+6+1 | |||||||||||
| 7 | 1 | 7 | 21 | 35 | 35 | 21 | 7 | 1 | 1+7+21+35+35+21+7+1 | ||||||||||
| 8 | 1 | 8 | 28 | 56 | 70 | 56 | 28 | 8 | 1 | 1+8+28+56+70+56+28+8+1 | |||||||||
The dimension of the kth element is given by:
n!/(n-k)! k!
