definition. Grade involution [wieser2022formalizing] [ca-000I]

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CliffordAlgebra.involute

L∃∀N

Grade involution, intuitively, is negating each basis vector.

Formally, it's an algebra homomorphism \(\alpha : \mathcal {C}\kern -2pt\ell (Q) \to _{a} \mathcal {C}\kern -2pt\ell (Q)\), satisfying:

1. \(\alpha \circ \alpha = \operatorname {id}\)
2. \(\alpha (\iota (m)) = - \iota (m)\)

for all \(m \in M\).

That is, the following diagram commutes:

It's called main involution \(\alpha \) or main automorphism in [jadczyk2019notes], the canonical automorphism in [gallier2008clifford].

It's denoted \(\hat {m}\) in [lounesto2001clifford], \(\alpha (m)\) in [jadczyk2019notes], \(m^*\) in [chisolm2012geometric].