## Defintion ## Given a [[commutative ring]] $R$, a $R$-**geometric algebra** is an $R$-[[algebra (ring theory)|algebra]] $A$ with a canonical ring homomorphism $\iota: R \to A$, with a binary function $\langle (-) \rangle_{(-)}: A \times \mathbb{N} \to A$ called the **grade projection operator** such that $$\prod_{a:A} a = \sum_{n:\mathbb{N}} \langle a \rangle_n$$ $$\prod_{a:A} \prod_{b:A} \prod_{n:\mathbb{N}} \langle a + b \rangle_n = \langle a \rangle_n + \langle b \rangle_n$$ $$\prod_{a:A} \prod_{c:A} \prod_{n:\mathbb{N}} (c = \langle c \rangle_0) \times (\langle c a \rangle_n = c \langle a \rangle_n)$$ $$\prod_{a:A} \prod_{n:\mathbb{N}} \langle \langle a \rangle_n \rangle_n = \langle a \rangle_n$$ $$\prod_{a:A} \prod_{m:\mathbb{N}} \prod_{n:\mathbb{N}} (m \neq n) \times (\langle \langle a \rangle_m \rangle_n = 0)$$ For a natural number $n:\mathbb{N}$, the image of $\langle (-) \rangle_n$ under $A$ is called the **$n$-vector space** and is denoted as $\langle A \rangle_n$. $$\langle A \rangle_0 \cong R$$ $$\prod_{v:\langle A \rangle_1} \Vert \sum_{r:\langle A \rangle_0} v^2 = r \Vert$$ Terms of $A$ are called **multivectors**. The terms of $\langle A \rangle_n$ are called **$n$-vectors**, $0$-vectors are called **scalars** and $1$-vectors are just called **vectors**. Every $R$-geometric algebra is a $R$-[[Clifford algebra]]. ## See also ## * [[Clifford algebra]] * [[algebra (ring theory)]] * [[real geometric algebra]] ## 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