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I'm trying to understand the proof of Proposition 1.5 in Spin Geometry by H. B. Lawson, JR. and M.-L. Michelsohn.


This says that if $V = V_1 \oplus V_2$ is an $q$-orthogonal decomposition of the vector space $V$ where $(V_i, q_i)$ are quadratic spaces with quadratic forms $q_i$ for $i=1,2$ and $q=q_1 \oplus q_2$. Then, there is a natural isomorphism of Clifford algebras $$Cl(V,q) \simeq Cl(V_1,q_1) \phantom{.} \hat{\otimes} \phantom{.} Cl(V_2,q_2).$$ where $\hat{\otimes}$ denotes the $\mathbb{Z}_2$-graded tensor of algebras.


They consider the map $f\colon V_1\bigoplus V_2 \rightarrow Cl(V_1,q_1) \phantom{.} \hat{\otimes} \phantom{.} Cl(V_2,q_2)$, such that $v_1+v_2$ is sent to $v_1\otimes 1 + 1\otimes v_2$, where $e_i \in V_i$ for $i=1,2$. Now $$f(v_1+v_2)^2=(v_1\otimes 1 + 1\otimes v_2)^2=v_1^2\otimes1 + 1\otimes v_2^2 + v_1\otimes v_2 \\ + (-1)^{\mathrm{deg}(v_1)\mathrm{deg}(v_2)} v_1\otimes v_2= -(q_1(v_1)+q_2(v_2)) 1\otimes 1. $$

I can not understand this last inequality since I don't see how can $v_1\otimes v_2 + (-1)^{\mathrm{deg}(v_1)\mathrm{deg}(v_2)} v_1\otimes v_2$ cancel.

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Elements of $V_1$ and $V_2$ always have degree $1$ in the Clifford algebra, so $\text{deg}(v_1) \text{deg}(v_2) = 1$.

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  • $\begingroup$ uff I didn't realise... $\endgroup$ – J. Salieri Sep 8 '19 at 21:48

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