\chapter{Coalgebras \& co}
Coalgebras, bialgebras, Hopf algebras, convolution

Dude, these are *cool*, and these are *everywhere*.

They are a deep, important, and overlooked structure.

Product of polynomials or power series is called "Cauchy product"
(on coefficients) and is a discrete convolution

is there a notion of a "self-dual" bialgebra/Hopf algebra?
  (i.e., with a map A -> A^*?) (hey, End V is self-dual!)
  actually, is every Hopf algebra self-dual?
  is the duality map the antipode etc???

what you need of finite dim is identification
  (A \otimes A)^* = A^* \otimes A^*
  [always a map, but in general * x * is a submodule of (A x A)^*]

Hopf algebras:
when you have an algebraic structure on a monoid, you get a bialgebra
when it's on a group, you get a Hopf algebra.

Lie Hopf algebra?
(or are these exclusive)

Hopf algebra & H-space: same guy



Lie bialgebra structure on the vector space generated by the free homotopy classes of curves on a surface
Goldman found that a Lie bracket is naturally defined using the intersection points of representatives and the usual loop product at these intersections. Later on, Turaev found a Lie coalgebra structure on the same vector space, using self-intersection points and loop coproduct.

Hodge star:
Hopf algebra!
(I think...)
cohomology always has an algebra structure;
if the Kunneth theorem holds (over fields and such),
homology has a coalgebra structure
(by law of conservation of ``co". (kidding!!))

oooooooooooooo!!!!!!!!!!!!
Ok,
you have a bialgebra (indeed, Hopf algebra)
*that is identified with its dual*
[EG, Z[G]]
So the bialgebra has a multiplication and comultiplication
...and there are also the *dual* operations.
  So you have 2 multiplications and 2 comultiplications!!
So:
- multiplication of functions is natural
- convolution of functions (the *other* multiplication) is dual to the comultiplication

[can't multiply distributions,
 but can convolve]


functions:
mult:
fg(x) = f(x)g(x)
comult:
\nabla f (x,y) = f(xy)

convolution:
  (coming from g*h)
fg(x) = \int_{s+t=x} f(s)g(t)
comult: ??
  (coming from g -> g x 1 + 1 x g)
\nabla f (x, y) = f(x) + f(y) ????

integrate 

Functions on a magma (with measure, so you can integrate)
have *two* natural multiplications:

X -> X x X -> X
pull back along diagonal;
integrate along fibers (anti-diagonals!)
convolution

Draw picture!
line, and all perpendiculars
viewing X as subset, or as quotient set
