Hopf theory MOC

Sweedler’s small Hopf algebra

Sweedler’s small Hopf algebra is a 4-dimensional noncommutative noncocommutative Hopf algebra with basis . hopf If is Sweedler’s large Hopf algebra, then

defines a Hopf ideal, and

is the Quotient Hopf algebra.

Representation theory

The simple modules for correspond to those for , namely

where one can show the composition series

The projective covers are

whence the Cartan matrix is

Proof

First we will show that and are indecomposable projective modules. Since

we see that are -modules and are thus projective modules since

Indecomposability follows from the fact that each of these have unique submodules

Now

so we see is indeed the projective cover of for . Moreover we have the composition series

for , giving the Cartan matrix above.

Dual

The dual to has the basis where are the characters of respectively and are thus grouplike, while and .

---

tidy | en | sembr