Pointed Hopf algebras, the Dixmier-Moeglin Equivalence and Noetherian group algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908613841256448 |
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| author | Bell, Jason P. Brown, Ken A. Stafford, J. Toby |
| author_facet | Bell, Jason P. Brown, Ken A. Stafford, J. Toby |
| contents | This paper addresses the interactions between three properties that a group algebra or more generally a pointed Hopf algebra may possess: being noetherian, having finite Gelfand-Kirillov dimension, and satisfying the Dixmier-Moeglin equivalence. First it is shown that the second and third of these properties are equivalent for group algebras $kG$ of polycyclic-by-finite groups, and are, in turn, equivalent to $G$ being nilpotent-by-finite. In characteristic $0$, this enables us to extend this equivalence to certain cocommutative Hopf algebras.
In sections 3 and 4 of the paper finiteness conditions for group algebras are studied. Thus in $§$3 we examine when a group algebra satisfies the Goldie conditions, while in the final section we discuss what can be said about a minimal counterexample to the conjecture that if $kG$ is noetherian then $G$ is polycyclic-by-finite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_23730 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pointed Hopf algebras, the Dixmier-Moeglin Equivalence and Noetherian group algebras Bell, Jason P. Brown, Ken A. Stafford, J. Toby Rings and Algebras Group Theory Quantum Algebra Representation Theory Primary 16R20, Secondary 16T20, 17B35 This paper addresses the interactions between three properties that a group algebra or more generally a pointed Hopf algebra may possess: being noetherian, having finite Gelfand-Kirillov dimension, and satisfying the Dixmier-Moeglin equivalence. First it is shown that the second and third of these properties are equivalent for group algebras $kG$ of polycyclic-by-finite groups, and are, in turn, equivalent to $G$ being nilpotent-by-finite. In characteristic $0$, this enables us to extend this equivalence to certain cocommutative Hopf algebras. In sections 3 and 4 of the paper finiteness conditions for group algebras are studied. Thus in $§$3 we examine when a group algebra satisfies the Goldie conditions, while in the final section we discuss what can be said about a minimal counterexample to the conjecture that if $kG$ is noetherian then $G$ is polycyclic-by-finite. |
| title | Pointed Hopf algebras, the Dixmier-Moeglin Equivalence and Noetherian group algebras |
| topic | Rings and Algebras Group Theory Quantum Algebra Representation Theory Primary 16R20, Secondary 16T20, 17B35 |
| url | https://arxiv.org/abs/2507.23730 |