nLab non-associative geometric algebra




Super-Algebra and Super-Geometry



A non-associative version of a geometric algebra.


Given a commutative ring RR, a non-associative geometric algebra is an \mathbb{N} -graded RR-module AA with a bilinear function ()():A×AA(-)(-):A \times A \to A and a ring isomorphism j:A 0Rj:\langle A \rangle_0 \cong R such that

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

  • the product of every 11-vector with itself is a 00-vector: for all aA 1a \in \langle A \rangle_1 there exists cA 0c \in \langle A \rangle_0 such that a 2=ca^2 = c.

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

See also

Created on May 10, 2022 at 21:42:08. See the history of this page for a list of all contributions to it.