# Tensor products and tensor algebras

1. Mar 20, 2007

### asub

Hi all,

What is a good introductory book on tensor products and tensor algebras? For motivation, I have found Tom Coates's http://www.math.harvard.edu/~tomc/math25/tensor.pdf" [Broken] to be quite helpful, but I would like to do see some examples and do problems to understand it more thoroughly.

My main aim is to be able to construct Clifford algebras by taking quotient of tensor algebras. I have become able to construct Clifford algebras using exterior products (the way Clifford did it) and quadratic spaces. But understanding tensor algebras and using them to create Clifford algebras seems impregnable.

Thanks.

Last edited by a moderator: May 2, 2017
2. Mar 20, 2007

### George Jones

Staff Emeritus
Let $V$ be a finite-dimensional vector space over $\mathbb{R}$ and $g: V \times V \rightarrow \mathbb{R}$ be a non-degenerate bilinear form. Form the the tensor algebra

[tex] T = \mathbb{R} \oplus V \oplus V \otimes V \oplus V \oplus V \otimes V \otimes V \oplus ...[/itex]

and generate an ideal $I$ from $g \left( v , v \right) - v \otimes v$. Then, the universal Clifford algebra is $T/I.$

Try the second edition (1978) of Multilinear Algebra by Greub.

Last edited: Mar 20, 2007