algebras

Can someone comment on what you gain by choosing Clifford Algebras over Exterior Algebras?

Scalars + bivectors are isomorphic to the complex numbers provided the base field is R and the isomorphism is as vector spaces not as algebras or rings.

Various algebras can be super-helpful, for example the patch-commutation algebra in the Darcs version control system, and the relational algebra underlying SQL.

So something ought to be said about the peculiarity of talking about associative algebras while not having an operation that is closed.

I'm given to understand physics makes heavy use of Lie algebras, somewhere, if that makes a difference.

Would love to read Scott Aaronson on Clifford Algebras (and computation, of course).

It is not even clear that other popular names have precedent, let alone especially useful expressive power beyond what normal relational algebras afford.

Normed Division Algebras are a fascinating and worthwhile subject to look into.

I have no idea how important HoTT is, but I feel that FP and abstract algebras are merging into one.

E.g., I was the only student who showed that there are no countably infinite sigma algebras.

Monads are one example of an algebra for doing this; applicatives, effects and arrows are other algebras we could use instead.

This would decrease the utility of overloading for implementing some algebras.

Note that we can also manipulate boolean values using algebras other than "boolean algebra"; eg.

To define an operation on two vectors and not point out that it isn't a vector while stating that the geometric algebra is different "from all other associative algebras" is misleading.

The post makes it appear that using cross product to express Faraday's law is somehow related to Clifford Algebras.

I succeeded in grad school, by doing actual mathematical work (my PhD thesis was in Banach Algebras).