In the last lecture, Crista and I wondered about what the origin is of the term “connected” in Hopf algebras. Our guess was that it came from topology.
Here is why I thought that the term might have originated from topology: When one studies singular homology in algebraic topology, one of the first results one can prove about the singular homology group is that if is non-empty and path-connected, then Informally, our space comes in a single piece.
However, recall that we say that a Hopf algebra is connected if Note that is a field, and we know that there are fields that are uncountable. Hence, this notion of connectedness in algebra need not agree with the notion of connectedness in topology. (For instance, consider a graded Hopf algebra that has, say, a -bialgebra structure.) Additionally, is not a field.
This being said, then, does anyone know or suspect what the origin of the term “connected” might be?
For those of you interested in reading a popular reference in algebraic topology, professor Allen Hatcher from Cornell University has made freely available his textbook in the subject: http://www.math.cornell.edu/~hatcher/AT/ATpage.html. In particular, the result I am referring to can be found on page 109 of this textbook.