For a quasi-Hopf algebra H, we study two types of 1-cycle deformations for a coalgebra C within the category of Yetter-Drinfeld modules over H, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${}_H^H{\mathcal YD}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mi>H</mi> <mi>H</mi> </mmultiscripts> <mrow> <mi mathvariant="script">Y</mi> <mi>D</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation>. The two deformations produce C-comodule structures in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${}_H^H{\mathcal YD}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mi>H</mi> <mi>H</mi> </mmultiscripts> <mrow> <mi mathvariant="script">Y</mi> <mi>D</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation> and new coalgebra structures on C in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${}_H^H{\mathcal YD}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mi>H</mi> <mi>H</mi> </mmultiscripts> <mrow> <mi mathvariant="script">Y</mi> <mi>D</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation>, respectively. We show that the isomorphism types of these structures are described by a 1-homology <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\mathcal H}^1_H(C, H_0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mrow> <mi mathvariant="script">H</mi> </mrow> <mi>H</mi> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>C</mi> <mo>,</mo> <msub> <mi>H</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> that we will introduce. Then we apply our results to the so called symplectic fermion quasi-Hopf algebras, algebras recently introduced by Farsad, Gainutdinov and Runkel.

1-Cycle Deformations for Yetter-Drinfeld Coalgebras

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2026年3月11日

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Daniel Bulacu,
Blas Torrecillas

摘要

For a quasi-Hopf algebra H, we study two types of 1-cycle deformations for a coalgebra C within the category of Yetter-Drinfeld modules over H, ${}_H^H{\mathcal YD}$ $_{H}^{H} Y D$ . The two deformations produce C-comodule structures in ${}_H^H{\mathcal YD}$ $_{H}^{H} Y D$ and new coalgebra structures on C in ${}_H^H{\mathcal YD}$ $_{H}^{H} Y D$ , respectively. We show that the isomorphism types of these structures are described by a 1-homology ${\mathcal H}^1_H(C, H_0)$ $H_{H}^{1} (C, H_{0})$ that we will introduce. Then we apply our results to the so called symplectic fermion quasi-Hopf algebras, algebras recently introduced by Farsad, Gainutdinov and Runkel.