$H$-Equivariant Morita equivalences of Loewy-graded comodule algebras
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908586391633920 |
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| author | Van Grinsven, Jacob |
| author_facet | Van Grinsven, Jacob |
| contents | Let $H$ be a coradically graded Hopf algebra. For every Loewy-graded exact $H$-comodule algebra $A=\oplus_{n\geq 0} A(n)$ and $H_0$-equivariant Morita equivalence $A(0)\simeq_{H_0} X$, there exists a Loewy-graded $H$-comodule algebra $B$ (isomorphic to $X$ in degree zero) realizing an $H$-equivariant Morita equivalence $A\simeq_H B$. In addition, if every exact $H_0$-comodule algebra is $H_0$-equivariant Morita equivalent to a coideal subalgebra of $H_0$, 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 $H_0=\mathcal{KP}$, the Kac-Paljutkin Hopf algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_09540 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $H$-Equivariant Morita equivalences of Loewy-graded comodule algebras Van Grinsven, Jacob Quantum Algebra Representation Theory Let $H$ be a coradically graded Hopf algebra. For every Loewy-graded exact $H$-comodule algebra $A=\oplus_{n\geq 0} A(n)$ and $H_0$-equivariant Morita equivalence $A(0)\simeq_{H_0} X$, there exists a Loewy-graded $H$-comodule algebra $B$ (isomorphic to $X$ in degree zero) realizing an $H$-equivariant Morita equivalence $A\simeq_H B$. In addition, if every exact $H_0$-comodule algebra is $H_0$-equivariant Morita equivalent to a coideal subalgebra of $H_0$, 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 $H_0=\mathcal{KP}$, the Kac-Paljutkin Hopf algebra. |
| title | $H$-Equivariant Morita equivalences of Loewy-graded comodule algebras |
| topic | Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2510.09540 |