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Defintion

Given a commutative ring RR, a RR-geometric algebra is an aRRgraded $R$-module - and an algebra $R$-algebra AA with a canonical ring homomorphism ι i:R→A \iota: i:R R \to A , with a an binary function⟨(−)⟩ (−):A×ℕ→A\langle (-) \rangle_{(-)}: A \times \mathbb{N} \to Aisomorphism? called thegrade projection operatorj:⟨A⟩ 0≅Rj:\langle A \rangle_0 \cong R such and that aquadratic form (−) 2:⟨A⟩ 1→R(-)^2:\langle A \rangle_1 \to R.

∏ a:Aa=∑ n:ℕ⟨a⟩ n\prod_{a:A} a = \sum_{n:\mathbb{N}} \langle a \rangle_n

Every RR-geometric algebra is a RR-Clifford algebra.

∏ a:A∏ b:A∏ n:ℕ⟨a+b⟩ n=⟨a⟩ n+⟨b⟩ n\prod_{a:A} \prod_{b:A} \prod_{n:\mathbb{N}} \langle a + b \rangle_n = \langle a \rangle_n + \langle b \rangle_n
∏ a:A∏ c:A∏ n:ℕ(c=⟨c⟩ 0)×(⟨ca⟩ n=c⟨a⟩ n)\prod_{a:A} \prod_{c:A} \prod_{n:\mathbb{N}} (c = \langle c \rangle_0) \times (\langle c a \rangle_n = c \langle a \rangle_n)
∏ a:A∏ n:ℕ⟨⟨a⟩ n⟩ n=⟨a⟩ n\prod_{a:A} \prod_{n:\mathbb{N}} \langle \langle a \rangle_n \rangle_n = \langle a \rangle_n
∏ a:A∏ m:ℕ∏ n:ℕ(m≠n)×(⟨⟨a⟩ m⟩ n=0)\prod_{a:A} \prod_{m:\mathbb{N}} \prod_{n:\mathbb{N}} (m \neq n) \times (\langle \langle a \rangle_m \rangle_n = 0)

For a natural number n:ℕn:\mathbb{N}, the image of ⟨(−)⟩ n\langle (-) \rangle_n under AA is called the nn-vector space and is denoted as ⟨A⟩ n\langle A \rangle_n.
⟨A⟩ 0≅R\langle A \rangle_0 \cong R
∏ v:⟨A⟩ 1‖∑ r:⟨A⟩ 0v 2=r‖\prod_{v:\langle A \rangle_1} \Vert \sum_{r:\langle A \rangle_0} v^2 = r \Vert

Terms of AA are called multivectors. The terms of ⟨A⟩ n\langle A \rangle_n are called nn-vectors, 00-vectors are called scalars and 11-vectors are just called vectors.

Every RR-geometric algebra is a RR-Clifford algebra.

See also

References

  • G. Aragón, J.L. Aragón, M.A. Rodríguez (1997), Clifford Algebras and Geometric Algebra, Advances in Applied Clifford Algebras Vol. 7 No. 2, pg 91–102, doi:10.1007/BF03041220, S2CID:120860757

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