Sweedler's small Hopf algebra

Sweedler's small Hopf algebra $𝐻 4$ is a 4-dimensional noncommutative noncocommutative Hopf algebra with basis ${1, 𝑔, 𝑥, 𝑥 𝑔}$ . #m/def/hopf If $𝐻$ is Sweedler's large Hopf algebra, then

𝐼 : = ⟨ 𝑥 2, 𝑔 2 - 1, 𝑔 𝑥 + 𝑥 𝑔 ⟩

defines a Hopf ideal, and

𝐻 4 = 𝐻 / 𝐼

is the Quotient Hopf algebra.

Representation theory

The simple modules for $𝐻 4$ correspond to those for $𝕂 C 2$ , namely

𝑆 0 = 𝕂 𝑣 0, 𝑔 𝑣 0 = 𝑣 0, 𝑥 𝑣 0 = 0; 𝑆 1 = 𝕂 𝑣 1, 𝑔 𝑣 1 = - 𝑣 1, 𝑥 𝑣 1 = 0 .

where one can show the composition series

[𝐻 4] = 2 [𝑆 0] + 2 [𝑆 1] .

The projective covers are

𝑃 0 = s p a n 𝕂 {1 + 𝑔, 𝑥 + 𝑥 𝑔}; 𝑃 1 = s p a n 𝕂 {1 - 𝑔, 𝑥 - 𝑥 𝑔}

whence the Cartan matrix is

[1111]

Proof

First we will show that $𝑃 0$ and $𝑃 1$ are indecomposable projective modules. Since

𝑔 {1 + 𝑔, 𝑥 + 𝑥 𝑔} = {1 + 𝑔, - 𝑥 - 𝑥 𝑔} \subseteq 𝑃 0, 𝑥 {1 + 𝑔, 𝑥 + 𝑥 𝑔} = {𝑥 + 𝑥 𝑔, 0} \subseteq 𝑃 0, 𝑔 {1 - 𝑔, 𝑥 - 𝑥 𝑔} = {- 1 + 𝑔, 𝑥 - 𝑥 𝑔} \subseteq 𝑃 1, 𝑥 {1 - 𝑔, 𝑥 - 𝑥 𝑔} = {𝑥 - 𝑥 𝑔, 0} \subseteq 𝑃 1;

we see that $𝑃 0, 𝑃 1$ are $𝐻 4$ -modules and are thus projective modules since

𝐻 4 ≅ 𝐻 4 𝑃 0 \oplus 𝑃 1

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

𝑆 1 ≅ 𝐻 4 ⟨ 𝑥 + 𝑥 𝑔 ⟩ \leq 𝐻 4 𝑃 0, 𝑆 0 ≅ 𝐻 4 ⟨ 𝑥 - 𝑥 𝑔 ⟩ \leq 𝐻 4 𝑃 1 .

Now

𝑆 0 ≅ 𝐻 4 𝑃 0 ⟨ 𝑥 + 𝑥 𝑔 ⟩, 𝑆 1 ≅ 𝐻 4 𝑃 1 ⟨ 𝑥 - 𝑥 𝑔 ⟩ .

so we see $𝑃 𝑖$ is indeed the projective cover of $𝑆 𝑖$ for $𝑖 \in {1, 2}$ . Moreover we have the composition series

[𝑃 𝑖] = [𝑆 0] + [𝑆 1]

for $𝑖 \in {1, 2}$ , giving the Cartan matrix above.

Dual

The dual to $𝐻 4$ has the basis ${𝜒 0, 𝜒 1, 𝑓 2, 𝑓 3}$ where $𝜒 0, 𝜒 1$ are the characters of $𝑆 0, 𝑆 1$ respectively and are thus grouplike, while $𝑓 2 = 1 𝑥 + 1 𝑥 𝑔$ and $𝑓 3 = 1 𝑥 - 1 𝑥 𝑔$ .

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