Show element is invertible in a Clifford algebra

**redsoxfan325** · November 22nd 2011, 04:16 PM

Here is the problem:

Let $F$ be a field and $V$ a quadratic space with an anisotropic form $q$ not representing $1$ (i.e. $q(v)\neq1$ for all $v\in V$ ). Show that if $a\in F$ and $v\in V$ are not both zero, then $a+v$ is an invertible element in the Clifford algebra $C(V,q)$ .

I'm not really sure how to start this problem. I don't fully understand Clifford algebras, so I don't even really know what invertible elements even look like. I understand the definition - $T(V)/I$ , where I is the ideal generated by $v\otimes v-q(v)$ - but I don't understand the resulting algebra (except when $q$ is a binary or nonzero unary form).

Any help is appreciated.

**NonCommAlg** · November 22nd 2011, 06:47 PM

Originally Posted by redsoxfan325

Here is the problem:

Let $F$ be a field and $V$ a quadratic space with an anisotropic form $q$ not representing $1$ . Show that if $a\in F$ and $v\in V$ are not both zero, then $a+v$ is an invertible element in the Clifford algebra $C(V,q)$ .

I'm not really sure how to start this problem. I don't fully understand Clifford algebras, so I don't even really know what invertible elements even look like. I understand the definition - $T(V)/I$ , where I is the ideal generated by $v\otimes v-q(v)$ - but I don't understand the resulting algebra (except when $q$ is a binary or nonzero unary form).

Any help is appreciated.