Following Richard Borcherds' questions 34110 and 61315, I'm looking for interesting examples of Hopf algebras for an introductory Hopf algebras graduate course.
Some of the examples I know are well-known:
- Examples related to groups and Lie algebras,
- Sweedler and Taft Hopf algebras,
- Drinfeld-Jimbo quantum groups.
(These examples can be found for example in Montgomery's book on Hopf algebras.)
Other interesting examples are Lusztig small quantum groups and the
generalized small quantum groups of Andruskiewitsch and Schneider.
Therefore I'm looking for interesting and non-standard examples maybe related to combinatorics or to other branches of mathematics.
Best Answer
Symmetric functions, quasisymmetric functions, Connes-Kreimer algebra are all examples of combinatorial Hopf algebras. There are many more.
The original example is the cohomology ring of a Lie group.
Also Steenrod algebra and the analogue for any cohomology theory.