What is a high rank tensor?

Question

Can someone please give me a good example of a rank 3, 2x2x2 or 3x3x3 tensor? Where are these forms arise from? Is a 4x3x3 tensor say, a pressure tensor on a 3D space in 4-dimensions? Are there any 2x2x2 metric tensors?

Answer 1

An algebra $A$ is a vector space equipped with a multiplication $m : A \otimes A \to A$, hence (at least when $A$ is finite-dimensional) a rank $3$ tensor in $A^{\ast} \otimes A^{\ast} \otimes A$. Examples include matrix algebras and Lie algebras (note that I am not requiring that $m$ is associative).

If you're one of those people who uses "tensor" to mean "tensor field," then consider for example the Lie bracket of vector fields, which is also a rank $3$ tensor; it's a section of $T^{\ast} \otimes T^{\ast} \otimes T$ where $T$ is the tangent bundle.

Answer 2

There are many questions here, but here's the answer to the first one. If $a,b$ are independent, then the following $2\times 2\times 2$ tensor has rank 3: $a\otimes a\otimes b + a\otimes b\otimes a + b\otimes a\otimes a$