= Clifford module
{c}
{title2=$c(v)^2=Q(v)I$}
For a <quadratic form> $Q$ on a <vector space> $V$, a Clifford module is a <module> over the <Clifford algebra> $\operatorname{Cl}(V,Q)$. Equivalently it is a <vector space> $S$ with a linear assignment $v\mapsto c(v)\in\operatorname{End}(S)$ satisfying
$$
c(v)^2=Q(v)I.
$$
Over a <field> of characteristic different from two, expanding $c(v+w)^2$ gives $c(v)c(w)+c(w)c(v)=2B(v,w)I$, with $B$ the <bilinear form> associated to $Q$. The <universal property> of the <Clifford algebra> extends these generator actions to an action of the whole <algebra over a field>.
For $Q(x,y)=x^2+y^2$ over $\mathbb R$, the matrices $c(e_1)=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ and $c(e_2)=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ both square to $I$ and anticommute. They make $\mathbb R^2$ a Clifford module. Such modules provide the algebraic setting for <spinors>; changing the sign convention for $Q$ changes the required generator squares.
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