{"title":"Deligne范畴与泛泛李超代数","authors":"I. Entova-Aizenbud, V. Serganova","doi":"10.17323/1609-4514-2021-21-3-507-565","DOIUrl":null,"url":null,"abstract":"We study stabilization of finite-dimensional representations of the periplectic Lie superalgebras $\\mathfrak{p}(n)$ as $n \\to \\infty$. \nThe paper gives a construction of the tensor category $Rep(\\underline{P})$, possessing nice universal properties among tensor categories over the category $\\mathtt{sVect}$ of finite-dimensional complex vector superspaces. \nFirst, it is the \"abelian envelope\" of the Deligne category corresponding to the periplectic Lie superalgebra, in the sense of arXiv:1511.07699. \nSecondly, given a tensor category $\\mathcal{C}$ over $\\mathtt{sVect}$, exact tensor functors $Rep(\\underline{P})\\longrightarrow \\mathcal{C}$ classify pairs $(X, \\omega)$ in $\\mathcal{C}$ where $\\omega: X \\otimes X \\to \\Pi \\mathbf{1}$ is a non-degenerate symmetric form and $X$ not annihilated by any Schur functor. \nThe category $Rep(\\underline{P})$ is constructed in two ways. The first construction is through an explicit limit of the tensor categories $Rep(\\mathfrak{p}(n))$ ($n\\geq 1$) under Duflo-Serganova functors. The second construction (inspired by P. Etingof) describes $Rep(\\underline{P})$ as the category of representations of a periplectic Lie supergroup in the Deligne category $\\mathtt{sVect} \\boxtimes Rep(\\underline{GL}_t)$. \nAn upcoming paper by the authors will give results on the abelian and tensor structure of $Rep(\\underline{P})$.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2018-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"17","resultStr":"{\"title\":\"Deligne Categories and the Periplectic Lie Superalgebra\",\"authors\":\"I. Entova-Aizenbud, V. Serganova\",\"doi\":\"10.17323/1609-4514-2021-21-3-507-565\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study stabilization of finite-dimensional representations of the periplectic Lie superalgebras $\\\\mathfrak{p}(n)$ as $n \\\\to \\\\infty$. \\nThe paper gives a construction of the tensor category $Rep(\\\\underline{P})$, possessing nice universal properties among tensor categories over the category $\\\\mathtt{sVect}$ of finite-dimensional complex vector superspaces. \\nFirst, it is the \\\"abelian envelope\\\" of the Deligne category corresponding to the periplectic Lie superalgebra, in the sense of arXiv:1511.07699. \\nSecondly, given a tensor category $\\\\mathcal{C}$ over $\\\\mathtt{sVect}$, exact tensor functors $Rep(\\\\underline{P})\\\\longrightarrow \\\\mathcal{C}$ classify pairs $(X, \\\\omega)$ in $\\\\mathcal{C}$ where $\\\\omega: X \\\\otimes X \\\\to \\\\Pi \\\\mathbf{1}$ is a non-degenerate symmetric form and $X$ not annihilated by any Schur functor. \\nThe category $Rep(\\\\underline{P})$ is constructed in two ways. The first construction is through an explicit limit of the tensor categories $Rep(\\\\mathfrak{p}(n))$ ($n\\\\geq 1$) under Duflo-Serganova functors. The second construction (inspired by P. Etingof) describes $Rep(\\\\underline{P})$ as the category of representations of a periplectic Lie supergroup in the Deligne category $\\\\mathtt{sVect} \\\\boxtimes Rep(\\\\underline{GL}_t)$. \\nAn upcoming paper by the authors will give results on the abelian and tensor structure of $Rep(\\\\underline{P})$.\",\"PeriodicalId\":54736,\"journal\":{\"name\":\"Moscow Mathematical Journal\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2018-07-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"17\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Moscow Mathematical Journal\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.17323/1609-4514-2021-21-3-507-565\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Moscow Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.17323/1609-4514-2021-21-3-507-565","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Deligne Categories and the Periplectic Lie Superalgebra
We study stabilization of finite-dimensional representations of the periplectic Lie superalgebras $\mathfrak{p}(n)$ as $n \to \infty$.
The paper gives a construction of the tensor category $Rep(\underline{P})$, possessing nice universal properties among tensor categories over the category $\mathtt{sVect}$ of finite-dimensional complex vector superspaces.
First, it is the "abelian envelope" of the Deligne category corresponding to the periplectic Lie superalgebra, in the sense of arXiv:1511.07699.
Secondly, given a tensor category $\mathcal{C}$ over $\mathtt{sVect}$, exact tensor functors $Rep(\underline{P})\longrightarrow \mathcal{C}$ classify pairs $(X, \omega)$ in $\mathcal{C}$ where $\omega: X \otimes X \to \Pi \mathbf{1}$ is a non-degenerate symmetric form and $X$ not annihilated by any Schur functor.
The category $Rep(\underline{P})$ is constructed in two ways. The first construction is through an explicit limit of the tensor categories $Rep(\mathfrak{p}(n))$ ($n\geq 1$) under Duflo-Serganova functors. The second construction (inspired by P. Etingof) describes $Rep(\underline{P})$ as the category of representations of a periplectic Lie supergroup in the Deligne category $\mathtt{sVect} \boxtimes Rep(\underline{GL}_t)$.
An upcoming paper by the authors will give results on the abelian and tensor structure of $Rep(\underline{P})$.
期刊介绍:
The Moscow Mathematical Journal (MMJ) is an international quarterly published (paper and electronic) by the Independent University of Moscow and the department of mathematics of the Higher School of Economics, and distributed by the American Mathematical Society. MMJ presents highest quality research and research-expository papers in mathematics from all over the world. Its purpose is to bring together different branches of our science and to achieve the broadest possible outlook on mathematics, characteristic of the Moscow mathematical school in general and of the Independent University of Moscow in particular.
An important specific trait of the journal is that it especially encourages research-expository papers, which must contain new important results and include detailed introductions, placing the achievements in the context of other studies and explaining the motivation behind the research. The aim is to make the articles — at least the formulation of the main results and their significance — understandable to a wide mathematical audience rather than to a narrow class of specialists.