Hopf theory MOC

Sweedler’s small Hopf algebra

Sweedler’s small Hopf algebra 𝐻4 is a 4-dimensional noncommutative noncocommutative Hopf algebra with basis {1,𝑔,𝑥,𝑥𝑔}. hopf If 𝐻 is Sweedler’s large Hopf algebra, then

𝐼:=⟨𝑥2,𝑔2−1,𝑔𝑥+𝑥𝑔⟩

defines a Hopf ideal, and

𝐻4=𝐻/𝐼

is the Quotient Hopf algebra.

Representation theory

The simple modules for 𝐻4 correspond to those for 𝕂ℤ2, namely

𝑆0=𝕂𝑣0,𝑔𝑣0=𝑣0,𝑥𝑣0=0;𝑆1=𝕂𝑣1,𝑔𝑣1=−𝑣1,𝑥𝑣1=0.

where one can show the composition series

[𝐻4]=2[𝑆0]+2[𝑆1].

The projective covers are

𝑃0=span𝕂⁡{1+𝑔,𝑥+𝑥𝑔};𝑃1=span𝕂⁡{1−𝑔,𝑥−𝑥𝑔}

whence the Cartan matrix is

[1111]

Proof

First we will show that 𝑃0 and 𝑃1 are indecomposable projective modules. Since

𝑔{1+𝑔,𝑥+𝑥𝑔}={1+𝑔,−𝑥−𝑥𝑔}⊆𝑃0,𝑥{1+𝑔,𝑥+𝑥𝑔}={𝑥+𝑥𝑔,0}⊆𝑃0,𝑔{1−𝑔,𝑥−𝑥𝑔}={−1+𝑔,𝑥−𝑥𝑔}⊆𝑃1,𝑥{1−𝑔,𝑥−𝑥𝑔}={𝑥−𝑥𝑔,0}⊆𝑃1;

we see that 𝑃0,𝑃1 are 𝐻4-modules and are thus projective modules since

𝐻4≅𝐻4𝑃0⊕𝑃1

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

𝑆1≅𝐻4⟨𝑥+𝑥𝑔⟩≤𝐻4𝑃0,𝑆0≅𝐻4⟨𝑥−𝑥𝑔⟩≤𝐻4𝑃1.

Now

𝑆0≅𝐻4𝑃0⟨𝑥+𝑥𝑔⟩,𝑆1≅𝐻4𝑃1⟨𝑥−𝑥𝑔⟩.

so we see 𝑃𝑖 is indeed the projective cover of 𝑆𝑖 for 𝑖 ∈{1,2}. Moreover we have the composition series

[𝑃𝑖]=[𝑆0]+[𝑆1]

for 𝑖 ∈{1,2}, giving the Cartan matrix above.

Dual

The dual to 𝐻4 has the basis {𝜒0,𝜒1,𝑓2,𝑓3} where 𝜒0,𝜒1 are the characters of 𝑆0,𝑆1 respectively and are thus grouplike, while 𝑓2 =1𝑥 +1𝑥𝑔 and 𝑓3 =1𝑥 −1𝑥𝑔.

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