{"title":"来自非交换代数变形的交换泊松代数","authors":"Alexander V. Mikhailov, Pol Vanhaecke","doi":"10.1007/s11005-024-01855-3","DOIUrl":null,"url":null,"abstract":"<div><p>It is well-known that a formal deformation of a commutative algebra <span>\\(\\mathcal {A}\\)</span> leads to a Poisson bracket on <span>\\(\\mathcal {A}\\)</span> and that the classical limit of a derivation on the deformation leads to a derivation on <span>\\(\\mathcal {A}\\)</span>, which is Hamiltonian with respect to the Poisson bracket. In this paper we present a generalization of it for formal deformations of an arbitrary noncommutative algebra <span>\\(\\mathcal {A}\\)</span>. The deformation leads in this case to a Poisson algebra structure on <span>\\(\\Pi (\\mathcal {A}){:}{=}Z(\\mathcal {A})\\times (\\mathcal {A}/Z(\\mathcal {A}))\\)</span> and to the structure of a <span>\\(\\Pi (\\mathcal {A})\\)</span>-Poisson module on <span>\\(\\mathcal {A}\\)</span>. The limiting derivations are then still derivations of <span>\\(\\mathcal {A}\\)</span>, but with the Hamiltonian belong to <span>\\(\\Pi (\\mathcal {A})\\)</span>, rather than to <span>\\(\\mathcal {A}\\)</span>. We illustrate our construction with several cases of formal deformations, coming from known quantum algebras, such as the ones associated with the nonabelian Volterra chains, Kontsevich integrable map, the quantum plane and the quantized Grassmann algebra.\n</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11005-024-01855-3.pdf","citationCount":"0","resultStr":"{\"title\":\"Commutative Poisson algebras from deformations of noncommutative algebras\",\"authors\":\"Alexander V. Mikhailov, Pol Vanhaecke\",\"doi\":\"10.1007/s11005-024-01855-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>It is well-known that a formal deformation of a commutative algebra <span>\\\\(\\\\mathcal {A}\\\\)</span> leads to a Poisson bracket on <span>\\\\(\\\\mathcal {A}\\\\)</span> and that the classical limit of a derivation on the deformation leads to a derivation on <span>\\\\(\\\\mathcal {A}\\\\)</span>, which is Hamiltonian with respect to the Poisson bracket. In this paper we present a generalization of it for formal deformations of an arbitrary noncommutative algebra <span>\\\\(\\\\mathcal {A}\\\\)</span>. The deformation leads in this case to a Poisson algebra structure on <span>\\\\(\\\\Pi (\\\\mathcal {A}){:}{=}Z(\\\\mathcal {A})\\\\times (\\\\mathcal {A}/Z(\\\\mathcal {A}))\\\\)</span> and to the structure of a <span>\\\\(\\\\Pi (\\\\mathcal {A})\\\\)</span>-Poisson module on <span>\\\\(\\\\mathcal {A}\\\\)</span>. The limiting derivations are then still derivations of <span>\\\\(\\\\mathcal {A}\\\\)</span>, but with the Hamiltonian belong to <span>\\\\(\\\\Pi (\\\\mathcal {A})\\\\)</span>, rather than to <span>\\\\(\\\\mathcal {A}\\\\)</span>. We illustrate our construction with several cases of formal deformations, coming from known quantum algebras, such as the ones associated with the nonabelian Volterra chains, Kontsevich integrable map, the quantum plane and the quantized Grassmann algebra.\\n</p></div>\",\"PeriodicalId\":685,\"journal\":{\"name\":\"Letters in Mathematical Physics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-08-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s11005-024-01855-3.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Letters in Mathematical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s11005-024-01855-3\",\"RegionNum\":3,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Letters in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s11005-024-01855-3","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
Commutative Poisson algebras from deformations of noncommutative algebras
It is well-known that a formal deformation of a commutative algebra \(\mathcal {A}\) leads to a Poisson bracket on \(\mathcal {A}\) and that the classical limit of a derivation on the deformation leads to a derivation on \(\mathcal {A}\), which is Hamiltonian with respect to the Poisson bracket. In this paper we present a generalization of it for formal deformations of an arbitrary noncommutative algebra \(\mathcal {A}\). The deformation leads in this case to a Poisson algebra structure on \(\Pi (\mathcal {A}){:}{=}Z(\mathcal {A})\times (\mathcal {A}/Z(\mathcal {A}))\) and to the structure of a \(\Pi (\mathcal {A})\)-Poisson module on \(\mathcal {A}\). The limiting derivations are then still derivations of \(\mathcal {A}\), but with the Hamiltonian belong to \(\Pi (\mathcal {A})\), rather than to \(\mathcal {A}\). We illustrate our construction with several cases of formal deformations, coming from known quantum algebras, such as the ones associated with the nonabelian Volterra chains, Kontsevich integrable map, the quantum plane and the quantized Grassmann algebra.
期刊介绍:
The aim of Letters in Mathematical Physics is to attract the community''s attention on important and original developments in the area of mathematical physics and contemporary theoretical physics. The journal publishes letters and longer research articles, occasionally also articles containing topical reviews. We are committed to both fast publication and careful refereeing. In addition, the journal offers important contributions to modern mathematics in fields which have a potential physical application, and important developments in theoretical physics which have potential mathematical impact.