definition. Clifford algebra [pr-spin] [ca-0002]

✍️source

Let \(M\) be a module over a commutative ring \(R\), equipped with a quadratic form \(Q: M \to R\).

A Clifford algebra over \(Q\) is \[ \mathcal {C}\kern -2pt\ell (Q) \equiv T(M)/I_Q \] where \(T(M)\) is the tensor algebra of \(M\), \(I_Q\) is the two-sided ideal generated from the set \[ \{ m \otimes m - Q(m) \mid m \in M \}. \]

We denote the canonical linear map \(M \to \mathcal {C}\kern -2pt\ell (Q)\) as \(\iota _Q\).