{"title":"\\sl(M|N)和osp(M|N)简单列超拉的((operatorname{ad}^{\\otimes 3}\\)表示中的\\(3\\)-分裂卡西米尔算子和沃格尔参数化","authors":"A. P. Isaev, A. A. Provorov","doi":"10.1134/S004057792410009X","DOIUrl":null,"url":null,"abstract":"<p> We find universal characteristic identities for the <span>\\(3\\)</span>-split Casimir operator in the representation <span>\\(\\operatorname{ad}^{\\otimes 3}\\)</span> of the <span>\\(osp(M|N)\\)</span> and <span>\\(sl(M|N)\\)</span> Lie superalgebras. Using these identities, we construct projectors onto the invariant subspaces of these representations and find universal formulas for their superdimensions. All the formulas are in accordance with the universal description of subrepresentations of the <span>\\(\\operatorname{ad}^{\\otimes 3}\\)</span> representation of simple basic Lie superalgebras in terms of the Vogel parameters. </p>","PeriodicalId":797,"journal":{"name":"Theoretical and Mathematical Physics","volume":"221 1","pages":"1726 - 1743"},"PeriodicalIF":1.0000,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"\\\\(3\\\\)-split Casimir operator of the \\\\(sl(M|N)\\\\) and \\\\(osp(M|N)\\\\) simple Lie superalgebras in the representation \\\\(\\\\operatorname{ad}^{\\\\otimes 3}\\\\) and the Vogel parameterization\",\"authors\":\"A. P. Isaev, A. A. Provorov\",\"doi\":\"10.1134/S004057792410009X\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p> We find universal characteristic identities for the <span>\\\\(3\\\\)</span>-split Casimir operator in the representation <span>\\\\(\\\\operatorname{ad}^{\\\\otimes 3}\\\\)</span> of the <span>\\\\(osp(M|N)\\\\)</span> and <span>\\\\(sl(M|N)\\\\)</span> Lie superalgebras. Using these identities, we construct projectors onto the invariant subspaces of these representations and find universal formulas for their superdimensions. All the formulas are in accordance with the universal description of subrepresentations of the <span>\\\\(\\\\operatorname{ad}^{\\\\otimes 3}\\\\)</span> representation of simple basic Lie superalgebras in terms of the Vogel parameters. </p>\",\"PeriodicalId\":797,\"journal\":{\"name\":\"Theoretical and Mathematical Physics\",\"volume\":\"221 1\",\"pages\":\"1726 - 1743\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-10-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Theoretical and Mathematical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://link.springer.com/article/10.1134/S004057792410009X\",\"RegionNum\":4,\"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":"Theoretical and Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1134/S004057792410009X","RegionNum":4,"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}
\(3\)-split Casimir operator of the \(sl(M|N)\) and \(osp(M|N)\) simple Lie superalgebras in the representation \(\operatorname{ad}^{\otimes 3}\) and the Vogel parameterization
We find universal characteristic identities for the \(3\)-split Casimir operator in the representation \(\operatorname{ad}^{\otimes 3}\) of the \(osp(M|N)\) and \(sl(M|N)\) Lie superalgebras. Using these identities, we construct projectors onto the invariant subspaces of these representations and find universal formulas for their superdimensions. All the formulas are in accordance with the universal description of subrepresentations of the \(\operatorname{ad}^{\otimes 3}\) representation of simple basic Lie superalgebras in terms of the Vogel parameters.
期刊介绍:
Theoretical and Mathematical Physics covers quantum field theory and theory of elementary particles, fundamental problems of nuclear physics, many-body problems and statistical physics, nonrelativistic quantum mechanics, and basic problems of gravitation theory. Articles report on current developments in theoretical physics as well as related mathematical problems.
Theoretical and Mathematical Physics is published in collaboration with the Steklov Mathematical Institute of the Russian Academy of Sciences.