I think the author is interested in clifford algebras, or more specifically geometric algebras, rather than division algebras.
division algebras tend to be quite boring (if they are finite then they are just a finite field; if they are finite dimensional over an algebraically closed field then they are just the field itself. I guess the quaternions are an interested example in the non-algebraically closed case. but I think if you are over something other than R you're really just talking about a field extension)
clifford algebras are a sort of generalization of the exterior algebra one would have encountered in differential geometry and other spaces.
in fact it could be considered a "quantization" of the exterior algebra. as in "quantum groups". which is an entirely different part of maths. but that's not what this article is about.
I think using the language of geometric algebras / clifford algebras in physics as this article does versus the more traditional language is just a matter of taste.
division algebras tend to be quite boring (if they are finite then they are just a finite field; if they are finite dimensional over an algebraically closed field then they are just the field itself. I guess the quaternions are an interested example in the non-algebraically closed case. but I think if you are over something other than R you're really just talking about a field extension)
clifford algebras are a sort of generalization of the exterior algebra one would have encountered in differential geometry and other spaces.
in fact it could be considered a "quantization" of the exterior algebra. as in "quantum groups". which is an entirely different part of maths. but that's not what this article is about.
I think using the language of geometric algebras / clifford algebras in physics as this article does versus the more traditional language is just a matter of taste.