I'm reading Spin Geometry by Lawson and Michelsohn. In general Clifford algebras are defined by $Cl(V,q)=\mathcal{T}(V)/\mathfrak{I}(V)$, where $\mathcal{T}(V)$ is the tensor algebra of $V$ and $\mathfrak{I}(V)$ the ideal generated by the elements $v\otimes v+q(V)1$ ($q$ is a quadratic form). For $V=\mathbb{R}^{r+s}$ it follows that $q(x)=x_1^2+…+x_r^2-x_{r+1}^2-…-x_{r+s}^2$ and we write $Cl_{r,s}$.
In the book was mentioned that $Cl_{1,0} \cong \mathbb{C}$ and $Cl_{0,1} \cong \mathbb{R}\oplus\mathbb{R}$. How do I see that these Clifford algebras have dimension two?
Thanks for your help.
Best Answer
With basis $x$ (for $\mathbb{R}$ over $\mathbb{R}$) the tensor algebra is the polynomial algebra $\bigoplus_{n\geq 0}\mathbb{R}^{\otimes n}=\mathbb{R}[x]$, but $x^2=-1$ or $x^2=1$ in the two cases you list, so we get $Cl_{1,0}=\mathbb{R}[x]/(x^2+1)=\mathbb{C}$ or $Cl_{0,1}=\mathbb{R}[x]/(x^2-1)=\mathbb{R}^2$