1D Clifford Algebra. You get reals (1) and 1D unit vectors (i). 1*1=1, i*i=1, i*1=i. Not very interesting.
2D Clifford Algebra. You get reals (1), two orthogonal unit vectors (i and j), and imaginaries or pseudovectors (i). 1*1=1, 1*i=i, i*i=-1, i*i=1, i*j=i etc.
3D Clifford Algebra. You get reals, three orthogonal unit vectors, the i, j and k of quaternions, and the unit volume element, whose symbol I can't remember.
4D Clifford Algebra, where the fourth dimension is time and doesn't quite behave like all the others. This is completely out of control, as far as I can remember.
Can someone who remembers all this better than I do, fill in the details? There was something terribly clever about expressing all Maxwell's equations in four symbols, or something. -- PeterHartley
The problem is that for each "dimensional" entry in the table above, there are several choices.
Cl(2) is actually isomorphic to the quaternions, not Cl(3), though Cl(3) has several sub-algebras isomorphic to the quaternions. Of the 1-d Clifford algebras, one is described above, and the other is isomorphic to the complex numbers.
Some general comments not yet revised, was originally on CliffordAlgebra.
The central definition in the algebra is multiplication. For three dimensions, this unifies the dot product and cross product of Gibbs vector theory into one operation.
Terms involving two basis vectors are also called bivectors.
The term which is the product of all the basis vectors forms a pseudoscalar.
In simple systems, the basis vectors will be orthogonal, but this need not be so.
For orthogonal systems, each basis vector will have an old style dot product of zero with all of the others.
This is where it gets interesting The dot product of any basis vector with itself can be either +1 or -1. This means that there are several different algebras at each value of n, depending on the choices made.
So, in the table above, at n = 4 we can have
Clifford (4) - all positive, four dimensional algebra.
Clifford (3,1) - 3 positive, one negative. This is one used in physics a lot.
Clifford (2,2) - 2 positive, 2 negative. This is called a balanced algebra.
Clifford (1,3) - 1 positive, 3 negative, also used in physics.