I am trying to work through pages 303-305 of Fulton and Harris and have ran into a problem. I will first give a little bit of setup to try and make this post self contained.
Let $V$ be an even dimensional complex vector space and $Q$ the standard non-degenerate symmetric bilinear form on $V$. We form the Clifford algebra $\mathcal{Cl}(V,Q)$. This algebra splits up into even and odd parts: $\mathcal{Cl}(V,Q)=\mathcal{Cl}(V,Q)^{\text{even}}\oplus\mathcal{Cl}(V,Q)^{\text{odd}}$, as does the exterior algebra: $\wedge^{\star}W=\wedge W^{\text{even}}\oplus\wedge W^{\text{odd}}$.
With this choice of dimension and bilinear form, $V$ splits up into maximal isotropic subspaces: $V=W\oplus W'$. Following Fulton and Harris I have then been able to show that we have an isomorphgism of algebras:
$$
\mathcal{Cl}(V,Q)\cong\text{End}\left(\wedge^{\star}W\right)
$$
Fulton and Harris claim that from this decomposition and the above isomorphism that we have:
$$
\mathcal{Cl}(V,Q)^{\text{even}}\cong\text{End}\left(\wedge W^{\text{even}})\oplus\text{End}(\wedge W^{\text{odd}}\right)
$$
and this is where I get stuck. I proceeded as follows:
$$
\mathcal{Cl}(V,Q)
\cong
\text{End}\left(\wedge^{\star}W\right)
\cong
\text{End}(\wedge W^{\text{even}}\oplus\wedge W^{\text{odd}})
\cong
\text{End}\left(\wedge W^{\text{even}})\oplus\text{End}(\wedge W^{\text{odd}}\right)
$$
This is obviously not correct, since it would imply that $\mathcal{Cl}(V,Q)\cong \mathcal{Cl}(V,Q)^{\text{even}}$. I would really appreciate it if anybody could point out where I've gone wrong.
EDIT.
Thanks to David Hill I have realised thast the identification:
$$
\text{End}(\wedge W^{\text{even}}\oplus\wedge W^{\text{odd}})
\cong
\text{End}(\wedge W^{\text{even}})\oplus\text{End}(\wedge W^{\text{odd}})
$$
was incorrect. This can be seen by either noting that $\text{dim}\text{End}(\wedge W^{\text{even}}\oplus\wedge W^{\text{odd}})=2^{2n}$, while $\text{dim}\text{End}(\wedge W^{\text{even}})\oplus\text{End}(\wedge W^{\text{odd}})=2^{2n-1}$, or by noting the existence of odd homomorphisms such as $f_{w}:\wedge^{\text{even}}W\to\wedge^{odd}W$, $u\mapsto wu$ for $w\in W$.
This means that so far I have only established the following isomorphism:
$$
\mathcal{Cl}(V,Q)^{\text{even}}\oplus\mathcal{Cl}(V,Q)^{\text{odd}}
\cong
\text{End}(\wedge W^{\text{even}}\oplus\wedge W^{\text{odd}})
$$
I still need to somehow "split up", $\text{End}(\wedge W^{\text{even}}\oplus\wedge W^{\text{odd}})$, and "project out", the odd part of the Cifford algebra, but I am not sure how to go about this. Fulton and Harris claim that this can be done by noting that $\mathcal{Cl}(W)^{\text{even}}$ respects the splitting $\wedge^{\star}W=\wedge W^{\text{even}}\oplus\wedge W^{\text{odd}}$, but I am not sure exactly what that means. Any pointers would be greatly appreciated.
Best Answer
Okay, I think I have figured this out. First of all, by decomposing an endomorphism into block digonal and block anti-diagonal parts one can show that for $n$-dimensional algebras $V_{1},V_{2}$:
$$ \text{End}(V_{1}\oplus V_{2}) \cong \text{End}(V_{1}) \oplus \text{End}(V_{2}) \oplus \text{Hom}(V_{1},V_{2}) \oplus \text{Hom}(V_{2},V_{1}) $$
and so:
$$ \mathcal{Cl}(V,Q)^{\text{even}} \oplus \mathcal{Cl}(V,Q)^{\text{odd}} \cong \left( \text{End}(\wedge^{\text{even}}W) \oplus \text{End}(\wedge^{\text{odd}}W) \right) \oplus \left( \text{Hom}(\wedge^{\text{even}}W,\wedge^{\text{odd}}W) \oplus \text{Hom}(\wedge^{\text{odd}}W,\wedge^{\text{even}}W) \right) $$
Let $\phi$ be the above isomorphism (it is given explicitly in Fulton and Harris). On can then show directly that $\phi$ respects this direct sum decomposition. Explicitly we have: $$ \phi|_{\mathcal{Cl}(V,Q)^{\text{even}}}:\mathcal{Cl}(V,Q)^{\text{even}}\to\text{End}(\wedge^{\text{even}}W) \oplus \text{End}(\wedge^{\text{odd}}W) $$ and $$ \phi|_{\mathcal{Cl}(V,Q)^{\text{odd}}}:\mathcal{Cl}(V,Q)^{\text{odd}}\to\text{Hom}(\wedge^{\text{even}}W,\wedge^{\text{odd}}W) \oplus \text{Hom}(\wedge^{\text{odd}}W,\wedge^{\text{even}}W) $$ And from here, since $\phi=\phi|_{\mathcal{Cl}(V,Q)^{\text{even}}}\oplus\phi|_{\mathcal{Cl}(V,Q)^{\text{odd}}}$, we see that both restrictions must be isomorphisms.
I think that my main difficulty with this problem was a findamental misunderstanding of the Endomorphism ring as pointed out by David Hill. Beyond that, working out an explicit example for $V=\mathbb{C}^{4}$ helped me realise that $\phi$ respects the $\mathbb{Z}_{2}$ grading of the Clifford algebra.