By Eckhard M.S. Hitzer, University of Fukui, Prof. Ryosuke Nagaoka (University of the Air, Japan)
Yes. Now I will introduce what is called geometric algebra or abbreviated GA to you. A very important fundamental work for this is the Ausdehnungslehre of Grassmann. And based as an application of the Ausdehnungslehre of Grassmann Clifford defined what is now said the Clifford geometric algebra. And today we will see how the Clifford geometric algebra is defined and especially look at the examples in two and three dimensions, where the complex numbers and the quaternions appear.
Next what is an algebra over the real field of real numbers. It is a linear space together with a bilinear map, so two elements (a,b) are mapped to a new element in the linear space.
Now we turn to look at the bilinear map, which is the geometric product of vectors, which was introduced by Clifford. For defining this we look at an orthonormal basis of the linear space R^n and we take two vectors of different index. And when we multiply them, the multiplication according to Clifford is antisymmetric. e_k times e_l equals minus e_l times e_k.
And if the index is the same, so if we multiply a vector with itself, the result is either plus one - which will be applied in the following - or minus one.
And also it is demanded that this product is an associative product of vectors.
Now if we look at the orthonormal basis of the plane, e.g. the Euclidean plane in this case, e_1 and e_2, than the basis of the whole algebra - as we will see soon ? is: the number one, the two vectors e_1 and e_2, and what we call the product of e_1 and e_2, the bivector e_12. Here you see the group multiplication table and you see that the squares of the two vectors are just one and the product of the vector e_1 with e_2 is this element e_12 and the product of e_12 with itself is minus one.
So in the geometric algebra R_2 we have first of all scalars which are proportional to one, vectors which are made by linear combinations of e_1 and e_2, and we have the bivectors, the products of e_1 e_2. And if we put the scalar and the bivector together, the square of the bivector is minus one, we get a sub-algebra which is isomorphic to the complex numbers. And the vectors themselves just form, as we see here, the Euclidean plane R upper index 2.
So next we turn to the geometric algebra of the three-dimensional space, our normal space we live in, which is indicated by R lower index 3. And the orthogonal basis of our space, the linear space is e_1, e_2 and e_3.
And then we have the products of e_2 and e_3 and we give it the label ?i, we have the product of e_3 and e_1, give it the label ?j. We have the product, which we had before in two dimensions, of e_1 and e_2 and now we give it the name ?k. And we have something new, which only exists in three dimensions the product of all three vectors e_1, e_2 and e_3, and we give it the name capital I. Now here you see all the multiplication rules for multiplying all these elements, calculated with the help of the geometric product. And you clearly see here in the lower line the multiplication rules, which are identical to the rules for the quaternions.
So here we have the elements of the geometric algebra of the three space listed according to first scalars proportional to one, then the three vectors, which form the linear space R^3, the three bivectors i,j,k and the volume trivector capital I. If we take together the even elements, the scalar and the three bivectors, we get a subalgebra, the even part of the algebra, which is isomorphic to the quaternions H.
So now you have received a brief summary of an introduction to geometric algebra. And the definition was made by Clifford by introducing a new product for vectors and he based his work on what Grassmann had defined for the product of vectors. And applying it to the plane, to two dimensions, we find a subalgebra, so we find the complex numbers as the even subalgebra. Applying it to three dimensions we find in the algebra of the three dimensional space that the even subalgebra is isomorphic to the quaternions, which is very useful to describe rotations - how Hamilton described rotations.
Soli Deo Gloria. Created by
Eckhard Hitzer (Fukui).
Last Modified 16 January 2004
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