Tensor products and tensor algebras

[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" 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.

 

[George Jones]

My main aim is to be able to construct Clifford algebras by taking quotient of tensor algebras.

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 $T/I.$

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.

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

 

[tacman]

Hi there,

A great introductory book on tensor products and tensor algebras is "Tensor Products and Exterior Algebras: Multilinear Algebra" by I.M. Gelfand and S.V. Fomin. This book covers the basics of tensor products, including definitions, properties, and examples, as well as the relationship between tensor products and exterior algebras. It also includes exercises and problems for practice and understanding.

To understand how tensor algebras can be used to construct Clifford algebras, I would also recommend "Clifford Algebras and Spinors" by Pertti Lounesto. This book covers the construction of Clifford algebras using tensor products and provides a thorough understanding of the relationship between the two.

I hope this helps and good luck with your studies!