{"title":"On the algebra generated by μ ¯ , ∂ ¯ , ∂ , μ \\overline{\\mu },\\overline{\\partial },\\partial ,\\mu","authors":"S. Auyeung, Jin-Cheng Guu, Jiahao Hu","doi":"10.1515/coma-2022-0149","DOIUrl":null,"url":null,"abstract":"Abstract In this note, we determine the structure of the associative algebra generated by the differential operators μ ¯ , ∂ ¯ , ∂ \\overline{\\mu },\\overline{\\partial },\\partial , and μ \\mu that act on complex-valued differential forms of almost complex manifolds. This is done by showing that it is the universal enveloping algebra of the graded Lie algebra generated by these operators and determining the structure of the corresponding graded Lie algebra. We then determine the cohomology of this graded Lie algebra with respect to its canonical inner differential [ d , − ] \\left[d,-] , as well as its cohomology with respect to all its inner differentials.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":" ","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2022-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Complex Manifolds","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/coma-2022-0149","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
Abstract In this note, we determine the structure of the associative algebra generated by the differential operators μ ¯ , ∂ ¯ , ∂ \overline{\mu },\overline{\partial },\partial , and μ \mu that act on complex-valued differential forms of almost complex manifolds. This is done by showing that it is the universal enveloping algebra of the graded Lie algebra generated by these operators and determining the structure of the corresponding graded Lie algebra. We then determine the cohomology of this graded Lie algebra with respect to its canonical inner differential [ d , − ] \left[d,-] , as well as its cohomology with respect to all its inner differentials.
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Complex Manifolds is devoted to the publication of results on these and related topics: Hermitian geometry, Kähler and hyperkähler geometry Calabi-Yau metrics, PDE''s on complex manifolds Generalized complex geometry Deformations of complex structures Twistor theory Geometric flows on complex manifolds Almost complex geometry Quaternionic geometry Geometric theory of analytic functions Holomorphic dynamics Several complex variables Dolbeault cohomology CR geometry.
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