One can figure out an equation involving α [A "twisting function" used to generate "twisted group algebras"] that guarantees this new product will be associative. In this case we call α a `2-cocycle'. If α satisfies a certain extra equation, the product [...] will also be commutative, and we call α a `stable 2-cocycle'. For example, the group algebra R[Z2] is isomorphic to a product of 2 copies of R, but we can twist it by a stable 2-cocyle to obtain the complex numbers. The group algebra R[Z22] is isomorphic to a product of 4 copies of R, but we can twist it by a 2-cocycle to obtain the quaternions. Similarly, the group algebra R[Z23] is a product of 8 copies of R, and what we have really done in this section is describe a function α that allows us to twist this group algebra to obtain the octonions. Since the octonions are nonassociative, this function is not a 2-cocycle. However, its coboundary is a `stable 3-cocycle', which allows one to define a new associator and braiding for the category of Z23-graded vector spaces, making it into a symmetric monoidal category. In this symmetric monoidal category, the octonions are a commutative monoid object. In less technical terms: this category provides a context in which the octonions are commutative and associative! So far this idea has just begun to be exploited.
If only I knew anything at all about group cohomology, I think I might have a chance at understanding this...
I found the lyrics to this song in someone's livejournal userinfo, but I can't remember whose, and I can't find any other information about the song.
I finished up the homework for HOT and submitted it. A fine assignment, although I'm sad that we're not going to get as far during the remaining weeks of class as I had hoped the class would.