what's the relationship of tensor and multivector

what's the relationship of multivector in geometric algebra and tensor? Is tensor a subset of multivector?

Solution 1:

Every multivector corresponds to a tensor, but not every tensor corresponds to a multivector.

The classic picture of a tensor is a linear map, acting on any number of vectors and dual vectors, to produce a scalar. Inspired by exterior and geometric algebra, you can say a tensor acts on any number of vector or dual vector blades.

For example, rotation operators can be viewed as tensors: let $\underline R(a)$ be some rotation, then there exists a tensor $R(a,b) = \underline R(a) \cdot b$. Because there is such a close correspondence, it's not uncommon to refer to the rotation map itself as a tensor.

You can extend the rotation map to act on blades, and therefore whole multivectors, through addition in the geometric algebra, but the rotation map itself isn't an element of the geometric algebra--it's not a multivector.

So, even in geometric algebra, we have need or use of linear maps, of general tensors.

How can you build a tensor from a general multivector? Through any function that is linear in all its arguments. A simple way is to consider a general multivector $M$. For the sake of an example, let's say $M$ is composed of blades grade-4 or less. Then you can build a tensor, $T$, such that

$$T(a, b, c, d) = \langle M abcd \rangle_0$$

which is linear in all its arguments $a, b, c, d$ and produces a scalar.

For this reason, you might refer to a multivector in general as a tensor, too, since there is such a close and obvious correspondence between a multivector and an associated linear map.

Solution 2:

These resources explaining how to represent multivectors as tensors might be helpful or interesting to you (or those with similar questions):

http://www2.ic.uff.br/~laffernandes/teaching/2013.1/topicos_ag/lecture_18%20-%20Tensor%20Representation.pdf

https://www.docdroid.net/uwfvUxE/tensor-representation-of-geometric-algebra.pdf.html