<p>We study the quantum double of a finite abelian group <i>G</i> twisted by a 3-cocycle and give a sufficient condition when such a twisted quantum double will be gauge equivalent to an ordinary quantum double of a finite group. Moreover, we will determine when a twisted quantum double of a cyclic group is genuine. As an application, we contribute to the classification of coradically graded finite-dimensional pointed coquasi-Hopf algebras over abelian groups. As a byproduct, we show that the Nichols algebras <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1639_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="136" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {B}(M_1\oplus M_2 \oplus M_3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mn>1</mn> </msub> <mo>⊕</mo> <msub> <mi>M</mi> <mn>2</mn> </msub> <mo>⊕</mo> <msub> <mi>M</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are infinite-dimensional where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1639_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="84" /> </InlineMediaObject> <EquationSource Format="TEX">$M_1,M_2,M_3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>M</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>M</mi> <mn>3</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> are three different simple Yetter-Drinfeld modules of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1639_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$D_8$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mn>8</mn> </msub> </math></EquationSource> </InlineEquation>.</p>

On gauge equivalence of twisted quantum doubles

期刊论文
发表日期：

2025年5月26日

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Bowen Li,
Gongxiang Liu

摘要

We study the quantum double of a finite abelian group G twisted by a 3-cocycle and give a sufficient condition when such a twisted quantum double will be gauge equivalent to an ordinary quantum double of a finite group. Moreover, we will determine when a twisted quantum double of a cyclic group is genuine. As an application, we contribute to the classification of coradically graded finite-dimensional pointed coquasi-Hopf algebras over abelian groups. As a byproduct, we show that the Nichols algebras $\mathcal {B}(M_1\oplus M_2 \oplus M_3)$ $B (M_{1} \oplus M_{2} \oplus M_{3})$ are infinite-dimensional where $M_1,M_2,M_3$ $M_{1}, M_{2}, M_{3}$ are three different simple Yetter-Drinfeld modules of $D_8$ $D_{8}$ .