Would a set of tensors be an algebraic group closed under some operation?

Question

Could a set of tensors be known as an algebraic group or why would that not have a group properties? The reason I'm asking is to understand different tensors.

Answer 1

The question to ask would be "a group under what operation?" In any reasonable setting (say, for vector spaces $V$ and $W$, both over a field $K$), the space $V\otimes W$ is an $additive$ group, since it is closed under addition and additive inverses. It is not in general, a $multiplicative$ group. How would you even define $(v\otimes w) \cdot (x\otimes y)$? What would be the multiplicative identity? Would such a structure be closed under multiplicative inverses?

Given a vector space $V$, there is such a thing as the $tensor$ $algebra$ $T(V)$. This is not a multiplicative group, but it does provide a framework for multiplying tensors, and contains a multiplicative identity. See http://en.wikipedia.org/wiki/Tensor_algebra