Homotopy Type Theory
geometric algebra > history (Rev #3)
Defintion
Given a commutative ring R R , a R R -geometric algebra is an R R -algebra A A with a canonical ring homomorphism ι : R → A \iota: R \to A , with a binary function ⟨ ( − ) ⟩ ( − ) : A × ℕ → A \langle (-) \rangle_{(-)}: A \times \mathbb{N} \to A called the grade projection operator such that
∏ a : A a = ∑ n : ℕ ⟨ a ⟩ n \prod_{a:A} a = \sum_{n:\mathbb{N}} \langle a \rangle_n ∏ 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 ) × ( ⟨ c a ⟩ 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 : ℕ n:\mathbb{N} , the image of ⟨ ( − ) ⟩ n \langle (-) \rangle_n under A A is called the n n -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 ⟩ 0 v 2 = r ‖ \prod_{v:\langle A \rangle_1} \Vert \sum_{r:\langle A \rangle_0} v^2 = r \Vert
Terms of A A are called multivectors . The terms of ⟨ A ⟩ n \langle A \rangle_n are called n n -vectors , 0 0 -vectors are called scalars and 1 1 -vectors are just called vectors .
Every R R -geometric algebra is a R R -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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