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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.08390 |
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| _version_ | 1866917892355784704 |
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| author | Nehme, Jonas |
| author_facet | Nehme, Jonas |
| contents | The periplectic Lie superalgebra $\mathfrak{p}(n)$ is one of the most mysterious and least understood simple classical Lie superalgebras with reductive even part. We approach the study of its finite dimensional representation theory in terms of Schur--Weyl duality. We provide an idempotent version of its centralizer, i.e. the super Brauer algebra. We use this to describe explicitly the endomorphism ring of a projective generator for $\mathfrak{p}(n)$ resembling the Khovanov algebra of [BS11a]. We also give a diagrammatic description of the translation functors from [BDE19] in terms of certain bimodules and study their effect on projective, standard, costandard and irreducible modules. These results will be used to classify irreducible summands in $V^{\otimes d}$, compute $\mathrm{Ext}^1$ between irreducible modules and show that $\mathfrak{p}(n)$-mod does not admit a Koszul grading. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_08390 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Khovanov algebras for the periplectic Lie superalgebras Nehme, Jonas Representation Theory Quantum Algebra 17B10 The periplectic Lie superalgebra $\mathfrak{p}(n)$ is one of the most mysterious and least understood simple classical Lie superalgebras with reductive even part. We approach the study of its finite dimensional representation theory in terms of Schur--Weyl duality. We provide an idempotent version of its centralizer, i.e. the super Brauer algebra. We use this to describe explicitly the endomorphism ring of a projective generator for $\mathfrak{p}(n)$ resembling the Khovanov algebra of [BS11a]. We also give a diagrammatic description of the translation functors from [BDE19] in terms of certain bimodules and study their effect on projective, standard, costandard and irreducible modules. These results will be used to classify irreducible summands in $V^{\otimes d}$, compute $\mathrm{Ext}^1$ between irreducible modules and show that $\mathfrak{p}(n)$-mod does not admit a Koszul grading. |
| title | Khovanov algebras for the periplectic Lie superalgebras |
| topic | Representation Theory Quantum Algebra 17B10 |
| url | https://arxiv.org/abs/2312.08390 |