Geometric Algebras

Question

As far as I can tell there are several extensions of linear algebra which can be used to do geometry on $\mathbb{R}^n.$ There are: the Clifford Algebra, the Grassman Algebra, the Exterior Algebra, Geometric Algebra, Hamilton Algebra, "Tensor Algebra" (not really sure if this is an algebra - but you can use tensor to do a lot of the preceding stuff I believe), differential geometry/forms, and perhaps others. Edit: It seems like there is also pre-Lie algebras (and maybe just Lie algebras in general?) as well.

It's not hard for me to go to Wikipedia and look at the definition of each, but this isn't very helpful. So how are each of these defined, what are the related concepts, and how do they all fit in together? Are there any that I have missed?

Thanks.

Answer 1

Jean-Louis Loday has written up a huge list of algebras arising in algebra, geometry, physics and many other areas, in his paper Encyclopedia of types of algebras 2010. Note that the name of the author there, "G. W. Zinbiel" is a reference to Leibniz algebras, written from right to left, refering to the "dual" operad, see here.

Answer 2

So how are each of these defined, what are the related concepts, and how do they all fit in together?

Of the structures you mentioned, there are really not that many in play, and they are not that complicated.

"Grassman algebra" is just another word for exterior algebra.

"Geometric algebra" is an alias or a Clifford algebra using a nondegenerate form.

Exterior algebras are a special case of a Clifford algebra (the form is the zero form.)

I don't know what you could mean by "Hamilton algebra" if it's not the quaternions, which can be looked upon as a Clifford algebra.

The tensor algebra is of a different nature than the ones above. It is "the biggest" algebra generated by a vector space, one from which we can extract special algebras as quotients. The Clifford algebras are one of the quotients that we can construct this way.

So you are talking mainly about these things: tensor algebra, Clifford algebra, exterior algebra (the rest being redundant, and I mention exterior algebra on its own even though it is a special case of a Clifford algebra.)

I'm not going to copy the definitions here for you, I just trust that now that you basically only need to study two definitions, your task will be much easier.

Are there any that I have missed?

Yeah, there are a bunch that will probably pour in.

The counterpart of Clifford algebras that use alternating forms are called Weyl algebras.

I don't know if you count Lie algebras as being relevant, but they are certainly a main character in geometry of differentiable manifolds.

There is also a more special type of Lie algebra called a Poisson algebra that is also very important for geometry of symplectic manifolds.

There are Jordan algebras, which are of a completely different flavor.

One should also mention $C^\ast$-algebras because of their application in physics.