Definition of Clifford Algebras

Question

I'm currently studying algebras, and in particular Clifford Algebras. Primarily I've been looking at this paper, and on page 7 and 8 the definition of a Clifford Algebra is given. I have a broad understanding of tensor algebras, but I'm struggling to understand the definition of Clifford algebra by the quotient algebra $T(V)$ with the ideal $I_q$ where $$I_q = < v \otimes v + q(x)1_{T(V)}>$$ In particular, it isn't clear to me why $I_q$ is a proper ideal, or how to interpret its elements, and further the elements of $T(V)/I_q$. I know this might be a lot of questions all in one, but any insights would be greatly appreciated!

Answer 1

To try to correct the notation a little, it is usually defined by $$I_q = \langle \{v \otimes v - q(v)1_{T(V)}\mid v\in V\}\rangle$$

Almost every resource on Clifford algebras demonstrates that if you select a basis $e_1, e_2,\ldots e_n$ for $V$, the collection of products of the form $e_{i_1}\cdot e_{i_2}\cdots e_{i_k}$ where $k\geq 0$ and $i_1<i_2<\ldots < i_k\leq n$ (including the empty one that produces $1$ and the ones that are just single elements of the basis) is a basis for the Clifford algebra. Within that argument is the evidence that the ideal is proper.

This also gives a concrete representation of the elements, as linear combinations of what they call "blades." It is reminiscent of the tensor algebra itself, except that now it's finite dimensional, of dimension $2^n$.

About interpretation in practice: for one thing, there are multiple interpretations of elements of the Clifford algebra. Some are used to represent what is acted on, some are used to represent what is acting. Depending on the context, one may be using an ordinary linear model, a homogenous coordinate model, or a "conformal model" or something else. The second thing is that I don't think all elements have interpretations.

Understanding the various ways to use and interpret the algebra elements is a somewhat extensive topic, but fortunately a lot of people have a lot of good texts that attempt to explain the schemes. I recommend starting with the wiki and then moving on to the numerous open source resources available online.