# Clifford algebra isomorphic to tensor algebra or exterior algebra?

1. Dec 22, 2006

### precondition

Unfortunately there seems to be a misprint in the paper I'm reading which is an introduction to clifford algebra, it says:(I highlighted in red possible misprint, either one of them has to be true misprint if you know what I mean)

The Clifford algebra C(V) is isomorphic to the tensor algebra Lambda(V) and is therefore a 2^{dim(V)} dimensional vector space with generators blah blah blah...

Now, I know C(V) is defined as T(V)/I with you know what "I" so I'm wondering how can there be isomorphism between C(V) and T(V) but on the other hand dimension 2^{dim(V)} is indeed dimension of tensor algebra right? Also the author said tensor algebra but then wrote Lambda...-_-

I'm confused~~

Last edited: Dec 22, 2006
2. Dec 22, 2006

### Hurkyl

Staff Emeritus
One of the names for $\Lambda(V)$ is the "antisymmetric tensor algebra (over V)".

Incidentally, while they are always isomorphic as vector spaces, I think they are only isomorphic as algebras when the Clifford algebra is built from the zero quadratic form.

3. Dec 23, 2006

### George Jones

Staff Emeritus
Yes.

Somtimes the vector space isomorphism between $Cl(V)$ and $\Lambda(V)$ is exploited by defining a second product on (vector space) $\Lambda(V)$ that makes (vector space) $\Lambda(V)$ with new product isomorphic to $Cl(V).$