The tensor algebra

We first use the tensor product to generate an algebra from an \({n}\)-dimensional vector space \({V}\). The \({k^{\textrm{th}}}\) tensor power of \({V}\), denoted \({T^{k}V}\), is the tensor product of \({V}\) with itself \({k}\) times; it forms a vector space of dimension \({n^{k}}\). The tensor algebra of \({V}\) is then defined to be the infinite direct sum of every tensor power:

\(\displaystyle TV\equiv\sum T^{k}V=\mathbb{R}\oplus V\oplus(V\otimes V)\oplus(V\otimes V\otimes V)\oplus\dotsb\)

(however, see the last paragraph of the section on Tensors).

The vector multiplication operation is \({\otimes}\), and thus the infinite-dimensional tensor algebra is associative. In fact, the tensor algebra can alternatively be defined as the free associative algebra on \({V}\), with juxtaposition indicated by the tensor product.

If a pseudo inner product, i.e. a nondegenerate symmetric bilinear form, is defined on \({V}\), it can be naturally extended to any \({T^{k}V}\) by extending the pairwise operation defined previously: if \({A=v_{1}\otimes v_{2}\otimes\dotsb\otimes v_{k},}\) and \({B=w_{1}\otimes w_{2}\otimes\dotsb\otimes w_{k}}\), we define

\(\displaystyle \left\langle A,B\right\rangle \equiv\prod\left\langle v_{i},w_{i}\right\rangle .\)

The pseudo inner product can then be extended to a nondegenerate symmetric multilinear form on all of \({TV}\) by defining it to be zero between elements from different tensor powers.

An Illustrated Handbook