Spencer
Bloch's course on
Hopf Algebras (in progress)
Here
are the scans of my notes for a course on Hopf algebras taught by
Spencer
Bloch at the University of Chicago during the autumn quarter of 2007.
Lecture
1
(09/25/2007). Most of this lecture was devoted to a discussion of
graded Hopf algebras over a field and two important examples of graded
Hopf algebras (one arising from rooted forests, and another
one
from finite graphs).
Lecture
2
(09/27/2007). The lecture was devoted to some preliminary constructions
and results used in the proof of the Milnor-Moore theorem. In
particular, spaces of primitive and indecomposable elements
for a
Hopf algebra are studied here.
Lecture
3
(10/02/2007).
The first part of the lecture was devoted to a proof of the
Milnor-Moore theorem. The second part began a discussion of path spaces
and loop spaces (in particular, Moore's model of the path space of a
topological space).
I
don't have the notes for Lecture 4.
Lecture 5
(10/09/2007). The
lecture included a few general words about the goals of Chen theory and
a discussion of the bar complex for associative algebras and Hochschild
(co)homology.
Lecture 6
(10/11/2007) was about differential forms on path spaces and the
definition of iterated integrals.
Lecture 7
(10/16/2007). Spencer
corrected the definition of iterated integrals from the previous
lecture. The rest of the lecture was devoted to properties of iterated
integrals viewed as differential forms on a loop space of X based at a
given point of X, rather than as differential forms on the full path
space of X. (Some of the formulas become simpler when we work with the
loop space of X. Besides, ultimately we are interested in the loop
spaces, not the path space.)