Representations of copointed Hopf algebras arising from the tetrahedron rack

Pogorelsky, Barbara; Vay, Cristian

doi:10.1007/s11565-013-0197-5

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Remarks 171

The Verma Module M(σ −)1

Proof By Corollary 27 L(e +)1

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2016

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“…(iv) There are examples with M(ĝ, ˆ ) M(g, ). For instance, let R be the Nichols algebra considered in [15]. Then R n top is not necessarily trivial as Yetter-Drinfeld module.…”

Section: Remarks 17mentioning

confidence: 99%

“…By (15), V⊗V M(σ, +) ⊕ M(σ, −) where (1 ± σ)x (12) ⊗v belongs to the submodule of weight (σ, ±). The action map applied to these elements gives Hence VQ = N 0 ⊕ N 1 ⊕ N 2 .…”

Section: The Verma Module M(σ −)mentioning

confidence: 99%

“…The proof of the next theorem and the description of Figure 4 are similar to the above subsection. Let E ⊂ M −1 (e, ρ) and C ⊂ M −2 (e, ρ) be the D(S 3 )-submodules of weight (σ, −) and (τ, 0) with basis (15); (σ, +)…”

Section: Proof By Corollary 27 L(e +)mentioning

confidence: 99%

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Verma and simple modules for quantum groups at non-abelian groups

Pogorelsky

Vay

2016

Advances in Mathematics

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The Drinfeld double D of the bosonization of a finite-dimensional Nichols algebra B(V) over a finite non-abelian group G is called a quantum group at a non-abelian group. We introduce Verma modules over such a quantum group D and prove that a Verma module has simple head and simple socle. This provides two bijective correspondences between the set of simple modules over D and the set of simple modules over the Drinfeld double D(G). As an example, we describe the lattice of submodules of the Verma modules over the quantum group at the symmetric group S 3 attached to the 12-dimensional Fomin-Kirillov algebra, computing all the simple modules and calculating their dimensions.

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“…(iv) There are examples with M(ĝ, ˆ ) M(g, ). For instance, let R be the Nichols algebra considered in [15]. Then R n top is not necessarily trivial as Yetter-Drinfeld module.…”

Section: Remarks 17mentioning

confidence: 99%

Section: The Verma Module M(σ −)mentioning

confidence: 99%