# Geometric Algebras

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.

• Yes, you have missed pre-Lie algebras, which arise in many areas of mathematics and physics - see a survey on them, and see Loday's book on "generalized Bialgebras and triples of operads", where he has a huge list of algebras, with definitions and properties. Mar 8, 2021 at 15:37

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.

• Ah, there is just too much math to learn. Mar 8, 2021 at 15:48
• For geometry one can focus a bit more, I suppose. Lie algebras, Clifford algebras, exterior algebras, Jordan algebras, pre-Lie algebras, alternative algebras, Poisson algebras, post-Lie algebras, ... Mar 8, 2021 at 15:52
• @Jbag1212 it is really not necessary to try to learn each of these one-by-one, unless that's something you want to do for it's own sake. They are all built out of linear/basic algebra, and are generally picked up while learning geometry (or other places they show up). Mar 9, 2021 at 4:35
• I agree, it is not necessary to learn all types of algebras. But it is useful to realize that mathematics is much less "narrow" than one might imagine at the beginning. So the question "Are there any that I have missed?" very often has the answer "Yes, you have missed many". Whether or not you can live without them really depends on your context. Mar 9, 2021 at 9:36

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.

• Did you mean to say " 'Geometric Algebra' is an alias of a Clifford algebra" ? Wikipedia says "Clifford's contribution was to define a new product, the geometric product, that united the Grassmann and Hamilton algebras into a single structure." en.wikipedia.org/wiki/Geometric_algebra Mar 8, 2021 at 16:31
• @Jbag1212 I think most people credit David Hestenes as re-kindling the usage of "geometric algebra" for what was then called Clifford algebras, yes. Since then "geometric algebra" has been used to push the study of finite dimensional real vector spaces with nondegenerate forms. My understanding is that Clifford algebraists study vector spaces over other fields, with arbitrary symmetric forms, and for infinite dimensional vector spaces. Mar 8, 2021 at 16:36
• My impression is that classically the theoretical side stuck to "clifford algebra" while the applications preferred to adopt the "geometric algebra" moniker. Mar 8, 2021 at 16:37
• @rschwieb Since you have mentioned Poisson algebras, I wanted to mention that pre and post Lie algebras are very important for the geometry of affinely flat manifolds (Milnor, Goldman) and Yang-Baxter equations. In this sense they are not necessarily less important than Poisson algebras. Of course, there might be different opinions about it, I know. Mar 9, 2021 at 9:44
• @DietrichBurde I saw you mention those, and I have to admit I'm completely ignorant of them! Life is too short to appreciate all these things :S Mar 9, 2021 at 14:10