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No, this is wrong. Geometric algebras aren't division algebras in general: they usually have zero divisors. Objects that live in a single grade are invertible,
by matt-noonan 5y ago
No, this is wrong. Geometric algebras aren't division algebras in general: they usually have zero divisors. Objects that live in a single grade are invertible, but composite objects don't always have multiplicative inverses.
As a concrete example, consider the elements 1 + x and 1 - x. Their product is 1 + x - x - xx = 1 + x - x - 1 = 0. So certainly 1 + x doesn't have an inverse, either.
- yodon 5y agoPresumably you also meant to say "when X^2 = 1"
- matt-noonan 5y agoYes, I meant that equation to be interpreted in the GA used in the article. But essentially all geometric algebras also have zero divisors, for similar reasons.