Hopf algebras are vector spaces but also rings (so they have dilation-by-scalars, “plus”, and “times”)

associativity is x⊗x⊗x ↦ 1⊗x⊗x → x⊗x ↦ 1⊗x → x and x⊗x⊗x ↦ x⊗x⊗1 → x⊗x ↦ x⊗1 → x

multiplication takes number ⊗ number → number, so comultiplication should take number → number ⊗ number

Hopf algebra is a vector space with multiplication, comultiplication, and antipodes.

Direct product

⊕ (=direct sum) is like two levers pulling on two things. They are sort-of-fake-ly wrapped together, but really the two levers are moving separate parts.

Tensor product

⊗ is really two levers moving one thing. The way to make it as un-complicated as possible is to think about moving only one of the levers at a time—and stipulating that, if they do move together, that should also be as decomposable as possible.

For example:

11 × 19   =   (10 + 1) × (20 − 1)

is the standard technique we’re suppose to teach children for thinking through hard multiplications. (Rather than make them commit a large lookup table to memory.) The reason the splitting above works is because of bilinearity—and with any other bilinear operator on other kinds of objects (like ⊗ with vectors, instead of multiplication with numbers), you can do something similar.

At minute 33, Federico Ardila shows how you can make up a “multiplication” even combining two different types of things. I think of sound as an ∞-dimensional wave (an infinite Fourier series looking, algebraically, roughly like a polynomial: it pairs a constant to each of 1 Hz, 2 Hz, 4 Hz, 8 Hz, … ad infinitum) — so what might a sound “times” a matrix be? Since the axiomatisation of ⊗ is pretty general, he can make this work. (You do end up with a hybrid object looking like color⊗matrix at the end, but the sounds and the matrices can intermix somewhat.

Polynomials over one letter, tensored with itself, gives a ring of polynomials over two letters

If you can tensor matrices, then you can tensor graphs, which in the Facebook era is easy to relate to real life. So 7⌫ + 2Δ + 3Γ (as graph-shapes) can now be meaningful.

permutations can be tensored

together. The coproduct looks pretty weird, like a Δ (abc) ≝ ∅⊗abc + a⊗bc + ab⊗c + abc⊗∅ style, and the product looks like shuffling.

In video 3 you’ll see a “pure algebraic” construction of the tensor of two vector spaces: it’s the free vector space (just an extremely dumb way to construct a big, generic object which is the highest-dimensionality vector-space you can get from a set), then quotient by the ideal generated by (a+b,c) − (a,c) − (b,c) & (a,b+c) − (a,b) − (a,c), and (λa,b) − λ(a,b) & (a,λb) − λ(a,b). In other words you take the biggest thing you can think of and divide out some stuff, in this case getting rid of FOIL leftovers.

The previous descriptions of tensors I’d seen were:

a 3-D matrix

something about stress on the faces of a cube

nasty index juggling (Christoffel symbol Γ)

Jacob Lurie

something about a braided monoidal category

There’s a big difference between setting up a system where ƒ(λa,λb) = λ ƒ(a,b) and one where ƒ(λa,b) = λ ƒ(a,b) = ƒ(a, λb), or ƒ(λa,a) = ƒ(a,λa) only when left matches right.