nLab non-associative geometric algebra

Idea

A non-associative version of a geometric algebra.

Definition

Given a commutative ring $R$, a non-associative geometric algebra is an $\mathbb{N}$-graded $R$-module $A$ with a bilinear function $(-)(-):A \times A \to A$ and a ring isomorphism $j:\langle A \rangle_0 \cong R$ such that

• for natural numbers $m:\mathbb{N}$ and $n:\mathbb{N}$, the product of every $m$-multivector and $n$-multivector is an $(m+n)$-multivector: for all $a \in \vert A \vert_m$ and $b \in \vert A \vert_n$, there exists $c \in \vert A \vert_{m+n}$ such that $a b = c$

• the product of every $1$-vector with itself is a $0$-vector: for all $a \in \langle A \rangle_1$ there exists $c \in \langle A \rangle_0$ such that $a^2 = c$.

Due to the ring isomorphism $j$ and the linearity of the grade projection operation, the algebra is unital.

See also

• geometric algebra

• N-graded module