Group-Graded Twisted Calabi--Yau Algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916508124315648 |
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| author | Baki, Yasmeen S. |
| author_facet | Baki, Yasmeen S. |
| contents | Historically, the study of graded (twisted or otherwise) Calabi--Yau algebras has meant the study of such algebras under an $\mathbb{N}$-grading. In this paper, we propose a suitable definition for a twisted $G$-graded Calabi-Yau algebra, for $G$ an arbitrary abelian group. Building on the work of Reyes and Rogalski, we show that a $G$-graded algebra is twisted Calabi-Yau if and only if it is $G$-graded twisted Calabi--Yau. In the second half of the paper, we prove that localizations of twisted Calabi--Yau algebras at elements which form both left and right denominator sets remain twisted Calabi--Yau. As such, we obtain a large class of $\mathbb{Z}$-graded twisted Calabi--Yau algebras arising as localizations of Artin-Schelter regular algebras. Throughout the paper, we survey a number of concrete examples of $G$-graded twisted Calabi--Yau algebras, including the Weyl algebras, families of generalized Weyl algebras, and universal enveloping algebras of finite dimensional Lie algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_09025 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Group-Graded Twisted Calabi--Yau Algebras Baki, Yasmeen S. Rings and Algebras Historically, the study of graded (twisted or otherwise) Calabi--Yau algebras has meant the study of such algebras under an $\mathbb{N}$-grading. In this paper, we propose a suitable definition for a twisted $G$-graded Calabi-Yau algebra, for $G$ an arbitrary abelian group. Building on the work of Reyes and Rogalski, we show that a $G$-graded algebra is twisted Calabi-Yau if and only if it is $G$-graded twisted Calabi--Yau. In the second half of the paper, we prove that localizations of twisted Calabi--Yau algebras at elements which form both left and right denominator sets remain twisted Calabi--Yau. As such, we obtain a large class of $\mathbb{Z}$-graded twisted Calabi--Yau algebras arising as localizations of Artin-Schelter regular algebras. Throughout the paper, we survey a number of concrete examples of $G$-graded twisted Calabi--Yau algebras, including the Weyl algebras, families of generalized Weyl algebras, and universal enveloping algebras of finite dimensional Lie algebras. |
| title | Group-Graded Twisted Calabi--Yau Algebras |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2405.09025 |