Let H be a coradically graded Hopf algebra. For every Loewy-graded exact H-comodule algebra <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$A=\oplus _{n\ge 0} A(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>=</mo> <msub> <mo>⊕</mo> <mrow> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$H_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>-equivariant Morita equivalence <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$A(0)\simeq _{H_0} X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <msub> <mo>≃</mo> <msub> <mi>H</mi> <mn>0</mn> </msub> </msub> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation>, there exists a Loewy-graded H-comodule algebra B (isomorphic to X in degree zero) realizing an H-equivariant Morita equivalence <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$A\simeq _H B$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <msub> <mo>≃</mo> <mi>H</mi> </msub> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation>. In addition, if every exact <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$H_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>-comodule algebra is <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$H_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>-equivariant Morita equivalent to a coideal subalgebra of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$H_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>, then every Loewy-graded exact H-comodule algebra is H-equivariant Morita equivalent to a coideal subalgebra of H. We also discuss Loewy-graded H-comodule algebras with <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$H_0={\mathcal{K}\mathcal{P}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mn>0</mn> </msub> <mo>=</mo> <mrow> <mi mathvariant="script">K</mi> <mi mathvariant="script">P</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation>, the Kac-Paljutkin Hopf algebra.

H-equivariant morita equivalences of loewy-graded comodule algebras

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2026年8月1日

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Jacob Van Grinsven

摘要

Let H be a coradically graded Hopf algebra. For every Loewy-graded exact H-comodule algebra $A=\oplus _{n\ge 0} A(n)$ $A = \oplus_{n \geq 0} A (n)$ and $H_0$ $H_{0}$ -equivariant Morita equivalence $A(0)\simeq _{H_0} X$ $A (0) ≃_{H_{0}} X$ , there exists a Loewy-graded H-comodule algebra B (isomorphic to X in degree zero) realizing an H-equivariant Morita equivalence $A\simeq _H B$ $A ≃_{H} B$ . In addition, if every exact $H_0$ $H_{0}$ -comodule algebra is $H_0$ $H_{0}$ -equivariant Morita equivalent to a coideal subalgebra of $H_0$ $H_{0}$ , then every Loewy-graded exact H-comodule algebra is H-equivariant Morita equivalent to a coideal subalgebra of H. We also discuss Loewy-graded H-comodule algebras with $H_0={\mathcal{K}\mathcal{P}}$ $H_{0} = K P$ , the Kac-Paljutkin Hopf algebra.