Clifford algebra

A Non-Commutative Quantum Observable Algebra is a Clifford Algebra

Definition 0.1   Let us briefly define the notion of a Clifford algebra. Thus, let us consider first a pair $ (V, Q)$ , where $ V$ denotes a real vector space and $ Q$ is a quadratic form on $ V$  . Then, the Clifford algebra associated to $ V$ , is denoted here as $ {\rm Cl}(V) = {\rm Cl}(V, Q)$ , is the algebra over $ \mathbb{R}$ generated by $ V$ , where for all $ v, w \in V$ , the relations: $ v \cdot w + w \cdot v = -2 Q(v,w)~,$ are satisfied; in particular, $ v^2 = -2Q(v,v)$  .

If $ W$ is an algebra and $ c : V {\longrightarrow}W$ is a linear map satisfying $ c(w) c(v) + c(v) c(w) = - 2Q (v, w)~, $ then there exists a unique algebra homomorphism $ \phi : {\rm Cl}(V) {\longrightarrow}W$ such that the diagram

$ \xymatrix{&&\hspace*{-1mm}{\rm Cl}(V)\ar[ddrr]^{\phi}&&\\ &&&&\\ \hspace{1mm} V \ar[uurr]^{{\rm Cl}}
\ar[rrrr]_{c}&&&& W\hspace{1mm}}$
commutes. (It is in this sense that $ {\rm Cl}(V)$ is considered to be `universal').



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