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(cf. LAWSON and MICHELSOHN's book on Spin Geometry page 8)

The book proves there is an natural embedding from a vector space $V$ to its Clifford algebra $Cl(V,q)$, where $q$ is a quadratic form on $V$. By definition, $Cl(V,q)$ is the quotient of the tensor algebra $\mathcal T(V)$ by $\mathcal I_q(V)$, the ideal generated by elements of form $v\otimes v +q(v) \cdot 1$.

But I cannot figure out details in its proof, especially the step by contraction with $q$. I discuss with my friends and also cannot write down it in a rigorous way.

Do you know how to show this embedding rigurously? The key point is to show it is injective. I will appreciate it if you can share your insights on this embedding $V \to Cl(V,q)$

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3 Answers 3

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It is somewhat "common knowledge" that the proof presented in the book is wrong. One way to show that $V$ injects into $Cl(V,q)$ is by considering representations. You can find the details in this thread.

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The embedding that you are looking for is very simple. Consider the natural projection $\pi : T(V)\to {\rm Cl}(V)$, where $T(V)$ is the tensor algebra and ${\rm Cl}(V)$ the Clifford algebra associated to $V$. Denote by $\phi : V\to T(V)$ the natural embedding of $V$ inside $T(V)$. Then $j:=\pi\circ \phi : V\to {\rm Cl}(V)$ is the desired map. In fact, $j$ is linear such that $j^{2}(u)=q(u)\cdot 1$, it is injective and the image $j(V)\subset {\rm Cl}(V)$ generates ${\rm Cl}(V)$ multiplicatively.

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$$T(V)\cong \oplus_{n\geq 0} V^{\otimes n}$$

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