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

Let $D 𝑛 (𝑞)$ be the Drinfel'd double of the Taft algebra. The non-isomorphic simple modules $𝑉 (ℓ, 𝑟)$ for $D 𝑛 (𝑞)$ are indexed by pairs $(ℓ, 𝑟)$ where $0 \leq ℓ < 𝑛$ is an integer and $𝑟 \in ℤ / 𝑛 ℤ$ . Here, $𝑉 (ℓ, 𝑟)$ is an $ℓ$ -dimensional $𝕂$ -space with basis ${𝑣 0, \dots, 𝑣 ℓ - 1}$ and $D 𝑛$ -action defined by¹ #m/thm/hopf

𝑎 𝑣 𝑖 = 𝑣 𝑖 + 1, 𝑑 𝑣 𝑖 = 𝛼 𝑖 (ℓ) 𝑣 𝑖 - 1, 𝑏 𝑣 𝑖 = 𝑞 𝑟 + 𝑖 𝑣 𝑖, 𝑐 𝑣 𝑖 = 𝑞 1 + 𝑖 - 𝑟 - ℓ 𝑣 𝑖

for $𝑖 = 1, \dots, ℓ$ , where $𝑣 - 1 = 𝑣 ℓ = 0$ and²

𝛼 𝑖 (ℓ) = (𝑖) 𝑞 (1 - 𝑞 𝑖 - ℓ) = (𝑞 𝑖 - 1) (1 - 𝑞 𝑖 - ℓ) 𝑞 - 1 .

Central character

Consider the central elements $𝑧, Ω \in Z (D 𝑛 (𝑞))$ . We have

𝑧 𝑉 (ℓ, 𝑟) = 𝑞 2 𝑟 + ℓ - 1; Ω 𝑉 (ℓ, 𝑟) = 𝑞 - 𝑟 + 𝑞 - 𝑟 - ℓ 𝑞 - 1 .

Calculation

For a basis vector $𝑣 𝑖$ we have

𝑧 𝑣 𝑖 = 𝑏 𝑐 - 1 𝑣 𝑖 = 𝑞 𝑟 + ℓ - 𝑖 - 1 𝑏 𝑣 𝑖 = 𝑞 2 𝑟 + ℓ - 1

and

Ω 𝑣 𝑖 = 𝑏 - 1 𝑎 𝑑 𝑣 𝑖 + 𝑏 - 1 𝑞 - 1 𝑣 𝑖 + 𝑞 - 1 𝑐 𝑞 - 1 𝑣 𝑖 = 𝛼 𝑖 (ℓ) 𝑏 - 1 𝑎 𝑣 𝑖 - 1 + 𝑞 - 𝑟 - 𝑖 + 𝑞 𝑖 - 𝑟 - ℓ 𝑞 - 1 𝑣 𝑖 = 𝛼 𝑖 (ℓ) 𝑏 - 1 𝑣 𝑖 + 𝑞 - 𝑟 - 𝑖 + 𝑞 𝑖 - 𝑟 - ℓ 𝑞 - 1 𝑣 𝑖 = (𝑞 𝑖 - 1) (1 - 𝑞 𝑖 - ℓ) 𝑞 - 𝑟 - 𝑖 + 𝑞 - 𝑟 - 𝑖 + 𝑞 𝑖 - 𝑟 - ℓ 𝑞 - 1 𝑣 𝑖 = 𝑞 - 𝑟 - 𝑞 𝑖 - 𝑟 - ℓ - 𝑞 - 𝑟 - 𝑖 + 𝑞 - 𝑟 - ℓ + 𝑞 - 𝑟 - 𝑖 + 𝑞 𝑖 - 𝑟 - ℓ 𝑞 - 1 𝑣 𝑖 = 𝑞 - 𝑟 (1 + 𝑞 - ℓ) 𝑞 - 1 𝑣 𝑖

as claimed.

Annihilators

Let $𝑃 ℓ, 𝑟 (𝑥)$ be the characteristic polynomial of $𝑏 𝑉 (ℓ, 𝑟) \in E n d 𝕂 𝑉 (ℓ, 𝑟)$ , so

𝑃 ℓ, 𝑟 (𝑥) = ℓ - 1 \prod 𝑖 = 0 (𝑥 - 𝑞 𝑟 + 𝑖) .

Since the eigenvalues of $𝑏 𝑉 (ℓ, 𝑟)$ are pairwise distinct, this is in fact the minimal polynomial of $𝑏 𝑉 (ℓ, 𝑟)$ . Then the ideal with presentation

𝐼 ℓ, 𝑟 = ⟨ 𝑃 ℓ, 𝑟 (𝑏), 𝑧 - 𝑞 2 𝑟 + ℓ - 1, Ω - 𝑞 - 𝑟 + 𝑞 - 𝑟 - ℓ 𝑞 - 1 ⟩

is the annihilator ideal of $𝑉 (ℓ, 𝑟)$ , and in fact

D 𝑛 (𝑞) / 𝐼 ℓ, 𝑟 ≅ E n d 𝕂 𝑉 (ℓ, 𝑟)

as $𝕂$ -monoids.

Proof

Let $A = D 𝑛 (𝑞) / 𝐼 ℓ, 𝑟$ , denote the corresponding congruence relation on $D 𝑛 (𝑞)$ by $(\equiv)$ and denote the congruence class of $𝑥 \in D 𝑛 (𝑞)$ as $[𝑥] = 𝑥 + 𝐼 ℓ, 𝑟 \in A$ . It is known by the Density theorem for $𝕂$ -monoids that the representation $D 𝑛 (𝑞) \to E n d 𝕂 𝑉 (ℓ, 𝑟)$ afforded by $𝑉 (ℓ, 𝑟)$ is surjective. By the Rank-nullity theorem, it therefore suffices to show that $d i m 𝕂 A \leq ℓ 2$ .

Consider the Lagrange polynomials

𝐿 𝑖 (𝑥) = \prod 0 \leq 𝑘 < ℓ 𝑘 \neq 𝑖 𝑥 - 𝑞 𝑟 + 𝑘 𝑞 𝑟 + 𝑖 - 𝑞 𝑟 + 𝑘 \in 𝕂 [𝑥], 𝑖 = 0, \dots, ℓ - 1

for the eigenvalues of $𝑏 𝑉$ , and let $𝑒 𝑖 : = 𝐿 𝑖 (𝑏)$ . Then $𝐿 𝑖 (𝑥) 2 - 𝐿 𝑖 (𝑥)$ annihilates all the eigenvalues of $𝑏 𝑉$ , since

𝐿 𝑖 (𝑞 𝑟 + 𝑗) 2 - 𝐿 𝑖 (𝑞 𝑟 + 𝑗) = 𝛿 2 𝑖 𝑗 - 𝛿 𝑖 𝑗 = 0,

so $𝑃 ℓ, 𝑟 (𝑥) ∣ 𝐿 𝑖 (𝑥) 2 - 𝐿 𝑖 (𝑥)$ and thus $𝑒 2 𝑖 \equiv 𝑒 𝑖$ . By ^P1 we have $\sum ℓ 𝑖 = 0 𝑒 𝑖 = 1$ , and clearly $𝑒 𝑖 𝑒 𝑗 + \equiv 𝛿 𝑖 𝑗$ . Therefore $𝑒 0, \dots, 𝑒 ℓ - 1$ form a complete system of orthogonal idempotents in $A$ . Now on the one hand, $𝑏 𝑒 𝑖 \equiv 𝑞 𝑟 + 𝑖 𝑒 𝑖$ since $(𝑏 - 𝑞 𝑟 + 𝑖) 𝑒 𝑖 \propto 𝑃 ℓ, 𝑟 (𝑏)$ , and on the other hand, if $𝑥 \in D 𝑛 (𝑞)$ such that $𝑏 𝑥 \equiv 𝑞 𝑟 + 𝑖 𝑥$ , then $𝑏 𝑒 𝑖 𝑥 \equiv 𝑏 𝑥$ and thus $𝑥 \equiv 𝑒 𝑖 𝑥$ . Thus we have

A = ℓ - 1 ⨁ 𝑖 = 0 𝑒 𝑖 A, 𝑒 𝑖 A = {𝑥 \in A : ∣ : 𝑏 𝑥 = 𝑞 𝑟 + 𝑖 𝑥} .

From this characterization, it is clear that $𝑎 𝑒 𝑖 \equiv 𝑒 𝑖 + 𝑗 𝑎 𝑒 𝑖$ and $𝑑 𝑒 𝑖 = 𝑒 𝑖 - 1 𝑑 𝑒 𝑖$ where we let $𝑒 - 1 = 𝑒 ℓ = 0$ . In fact,

A = ⟨ [𝑒 𝑖], [𝑎 𝑒 𝑖], [𝑑 𝑒 𝑖] ⟩ ℓ 𝑖 = 0

as a $𝕂$ -monoid, since

𝑎 = ℓ - 1 \sum 𝑖 = 0 𝑎 𝑒 𝑖, 𝑏 \equiv ℓ - 1 \sum 𝑖 = 0 𝑞 𝑟 + 𝑖 𝑒 𝑖, 𝑑 = ℓ - 1 \sum 𝑖 = 0 𝑑 𝑒 𝑖, 𝑐 \equiv 𝑞 1 - 2 𝑟 - ℓ 𝑏 .

Thus $A$ has a presentation as a quiver with relations, where the quiver is

c|https://q.uiver.app/#q=WzAsNCxbMCwwLCIwIl0sWzIsMCwiMSJdLFs0LDAsIlxcZWxsLTIiXSxbNiwwLCJcXGVsbC0xIl0sWzAsMSwiYWVfMCIsMCx7Im9mZnNldCI6LTF9XSxbMSwyLCIiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkb3R0ZWQifSwiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dLFsyLDMsImFlX3tcXGVsbC0yfSIsMCx7Im9mZnNldCI6LTF9XSxbMywyLCJkZV97XFxlbGwtMX0iLDAseyJvZmZzZXQiOi0xfV0sWzEsMCwiZGVfMSIsMCx7Im9mZnNldCI6LTF9XV0=

Applying the relation involving $Ω$ gives $[𝑑 𝑒 𝑖 + 1 𝑎 𝑒 𝑖] \propto [𝑒 𝑖]$ and $[𝑎 𝑒 𝑖 - 1 𝑑 𝑒 𝑖] \propto [𝑒 𝑖]$ . Thus any path from $𝑖$ to $𝑗$ in the above quiver yields the same element of $A$ up to scaling, so $d i m 𝕂 A \leq ℓ 2$ .

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

2000. Irreducible representations of a class of quantum doubles ↩
See $𝑞$ -analogue. ↩

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

Central character

Annihilators

Footnotes