Drinfel'd double of the Taft algebra

For $𝑛 \geq 2$ an integer, suppose $𝕂$ has a primitive $𝑛$ th root of unity $𝑞 \in 𝕂$ . The Drinfel'd double $D 𝑛 (𝑞)$ of the Taft algebra $H 𝑛 (𝑞)$ can be explicitly presented as the $𝑛 4$ -dimensional Hopf algebra generated as a $𝕂$ -monoid by ${𝑎, 𝑏, 𝑐, 𝑑}$ under relations #m/thm/hopf

𝑏 𝑎 = 𝑞 𝑎 𝑏, 𝑑 𝑏 = 𝑞 𝑏 𝑑, 𝑐 𝑎 = 𝑞 𝑎 𝑐, 𝑑 𝑐 = 𝑞 𝑐 𝑑, 𝑏 𝑐 = 𝑐 𝑏, 𝑑 𝑎 - 𝑞 𝑎 𝑑 = 1 - 𝑏 𝑐, 𝑎 𝑛 = 0 = 𝑑 𝑛, 𝑏 𝑛 = 1 = 𝑐 𝑛

with coproduct and coünit

Δ (𝑎) = 𝑎 \otimes 𝑏 + 1 \otimes 𝑎, Δ (𝑑) = 𝑑 \otimes 𝑐 + 1 \otimes 𝑑, Δ (𝑏) = 𝑏 \otimes 𝑏, Δ (𝑐) = 𝑐 \otimes 𝑐, 𝜖 (𝑎) = 0 = 𝜖 (𝑑), 𝜖 (𝑏) = 1 = 𝜖 (𝑐)

and antipous of order $2 𝑛$

𝜎 (𝑎) = - 𝑎 𝑏 - 1, 𝜎 (𝑏) = 𝑏 - 1, 𝜎 (𝑐) = 𝑐 - 1, 𝜎 (𝑑) = - 𝑑 𝑐 - 1

The standard basis is

{𝑎 𝑚 𝑏 𝑠 𝑐 𝑡 𝑑 ℓ : ∣ : 0 \leq 𝑚, 𝑠, 𝑡, ℓ < 𝑛} .

The $𝑛 2$ -dimensional $𝕂$ -submonoid generated by ${𝑎, 𝑏}$ is the Taft algebra $H 𝑛$ .¹

Quasitriangular structure

As a Drinfel'd double, $D 𝑛 (𝑞)$ is a Quasitriangular $K$ -bimonoid with universal $R$ -matrix²

𝑅 = 𝑛 - 1 \sum 𝑚, 𝑠, 𝑙 = 0 𝑎 𝑠 𝑏 𝑙 \otimes 𝑐 𝑚 𝑑 𝑠 𝑛 𝑞 𝑙 𝑚 (𝑠)! 𝑞 .

and thus corresponding tracelike

𝑢 = 𝑛 - 1 \sum 𝑚, 𝑠, 𝑙 = 0 𝜎 (𝑐 𝑚 𝑑 𝑠) 𝑎 𝑠 𝑏 𝑙 𝑛 𝑞 𝑙 𝑚 (𝑠)! 𝑞 = 𝑛 - 1 \sum 𝑚, 𝑠, 𝑙 = 0 (- 𝑑 𝑐 - 1) 𝑠 𝑐 - 𝑚 𝑎 𝑠 𝑏 𝑙 𝑛 𝑞 𝑙 𝑚 (𝑠)! 𝑞 = 𝑛 - 1 \sum 𝑚, 𝑠, 𝑙 = 0 (- 1) 𝑠 𝑞 𝑠 (𝑠 - 1) 2 - 𝑙 𝑚 𝑑 𝑠 𝑐 𝑠 - 𝑚 𝑎 𝑠 𝑏 𝑙 𝑛 (𝑠)! 𝑞

Central elements

Let

𝑧 : = 𝑏 𝑐 - 1, Ω : = 𝑏 - 1 𝑎 𝑑 + 𝑏 - 1 + 𝑞 - 1 𝑐 𝑞 - 1 .

$𝑧$ is central and grouplike.
$Ω$ is central.

Proof of 1

For ^C1, $𝑏 𝑐 - 1$ is grouplike as the product of grouplikes. We already have that $𝑏 𝑐 - 1$ commutes with $𝑏$ and $𝑐$ , it commutes with $𝑎$ since

𝑏 𝑐 - 1 𝑎 = 𝑏 𝑐 𝑛 - 1 𝑎 = 𝑞 𝑛 - 1 𝑏 𝑎 𝑐 𝑛 - 1 = 𝑞 𝑛 𝑎 𝑏 𝑐 - 1 = 𝑎 𝑏 𝑐 - 1

and $𝑑$ since

𝑑 𝑏 𝑐 - 1 = 𝑞 𝑏 𝑑 𝑐 𝑛 - 1 = 𝑞 𝑛 𝑏 𝑐 𝑛 - 1 𝑑 = 𝑏 𝑐 - 1 𝑑

Therefore $𝑏 𝑐 - 1$ is central, proving ^C1.

Observe that $[𝑏, 𝑏 - 1 𝑎 𝑑] = [𝑎, 𝑏 - 1 𝑎 𝑑] = 0$ . On the other generators we have

[𝑎, 𝑏 - 1 𝑎 𝑑] = 𝑎 𝑏 - 1 𝑎 𝑑 - 𝑏 - 1 𝑎 𝑑 𝑎 = 𝑞 - 1 𝑎 2 𝑏 - 1 𝑑 - 𝑞 - 1 𝑎 𝑏 - 1 (𝑞 𝑎 𝑑 + 1 - 𝑏 𝑐) = 𝑞 - 1 𝑎 2 𝑏 - 1 𝑑 - 𝑞 - 1 𝑎 2 𝑏 - 1 𝑑 - 𝑞 - 1 𝑎 𝑏 - 1 + 𝑞 - 1 𝑎 𝑐 = 𝑞 - 1 𝑎 (𝑐 - 𝑏 - 1); [𝑑, 𝑏 - 1 𝑎 𝑑] = 𝑑 𝑏 - 1 𝑎 𝑑 - 𝑏 - 1 𝑎 𝑑 2 = 𝑞 - 1 𝑏 - 1 𝑑 𝑎 𝑑 - 𝑏 - 1 𝑎 𝑑 2 = 𝑞 - 1 𝑏 - 1 (𝑞 𝑎 𝑑 + 1 - 𝑏 𝑐) 𝑑 - 𝑏 - 1 𝑎 𝑑 2 = 𝑏 - 1 𝑎 𝑑 2 + 𝑞 - 1 𝑏 - 1 𝑑 - 𝑞 - 1 𝑐 𝑑 - 𝑏 - 1 𝑎 𝑑 2 = 𝑞 - 1 (𝑏 - 1 - 𝑐) 𝑑 .

On the other hand,

[𝑎, 𝑏 - 1] = (1 - 𝑞 - 1) 𝑎 𝑏 - 1; [𝑎, 𝑐] = (1 - 𝑞) 𝑎 𝑐; [𝑑, 𝑏 - 1] = (𝑞 - 1 - 1) 𝑏 - 1 𝑑; [𝑑, 𝑐] = (𝑞 - 1) 𝑐 𝑑 .

It follows that

𝑏 - 1 𝑎 𝑑 + 𝑞 - 1 1 - 𝑞 - 1 𝑏 - 1 - 𝑞 - 1 1 - 𝑞 𝑐 = 𝑏 - 1 𝑎 𝑑 + 𝑏 - 1 + 𝑞 - 1 𝑐 𝑞 - 1

commutes with all the generators, and thus $Ω$ is central, proving ^C2.

Representation theory

Simple modules of the Drinfel'd double of the Taft algebra

Bibliography

#state/tidy | #lang/en | #SemBr

2000. Irreducible representations of a class of quantum doubles, §1. ↩
1999. A class of noncommutative and noncocommutative Hopf algebras: The quantum version, Theorem 3.6 ↩

Drinfel'd double of the Taft algebra

Quasitriangular structure

Central elements

Representation theory

Bibliography

Footnotes