Semisimplicity of theDS functor for the orthosymplectic Lie superalgebra.(English)Zbl 1497.17009
For a finite dimensional complex Lie superalgebra \(\mathfrak{g}\) and an odd nilpotent element \(x\), consider the cohomology of the complex \(V\stackrel{\rho(x)}{\longrightarrow} V\stackrel{\rho(x)}{\longrightarrow} V\) for \((V,\rho)\in \text{Rep}(\mathfrak{g})\). Duflo and Serganova defined a functor \(\mathrm{DS}_x: \text{Rep}(\mathfrak{g}) \to \text{Rep}(\mathfrak{g}_x)\), where \(\mathfrak{g}_x := \ker (\text{ad}(x)) / \text{im} (\text{ad}(x))\). For \(\mathfrak{osp}(m|2n)\), one has \(\mathfrak{g}_x \simeq \mathfrak{osp}(m-2k|2n-2k)\), where \(k\) is the rank of \(x\), so the DS-functor allows one to reduce questions about superdimensions to lower rank.
M. Gorelik and T. Heidersdorf prove that the Duflo-Serganova functor \(\mathrm{DS}_x\) attached to an odd nilpotent element \(x\) of \(\mathfrak{osp}(m|2n)\) sends a semisimple representation \(M\) of \(\mathfrak{osp}(m|2n)\) to a semisimple representation of \(\mathfrak{osp}(m-2k|2n-2k)\) where \(k\) is the rank of \(x\). They prove a closed formula for \(\mathrm{DS}_x(L(\lambda))\) in terms of the arc diagram attached to \(\lambda\).
M. Gorelik and T. Heidersdorf prove that the Duflo-Serganova functor \(\mathrm{DS}_x\) attached to an odd nilpotent element \(x\) of \(\mathfrak{osp}(m|2n)\) sends a semisimple representation \(M\) of \(\mathfrak{osp}(m|2n)\) to a semisimple representation of \(\mathfrak{osp}(m-2k|2n-2k)\) where \(k\) is the rank of \(x\). They prove a closed formula for \(\mathrm{DS}_x(L(\lambda))\) in terms of the arc diagram attached to \(\lambda\).
Reviewer: Mee Seong Im (Annapolis)
MSC:
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 17B20 | Simple, semisimple, reductive (super)algebras |
| 17B55 | Homological methods in Lie (super)algebras |
| 18M05 | Monoidal categories, symmetric monoidal categories |
| 18M25 | Tannakian categories |
Keywords:
representations of supergroups;Duflo-Serganova (DS) functor;superdimensions;orthosymplectic Lie superalgebra;orthosymplectic supergroup;symmetric monoidal functorCite
References:
| [1] | Brundan, J.; Stroppel, C., Highest weight categories arising from Khovanov’s diagram algebra. IV: The general linear supergroup, J. Eur. Math. Soc., 14, 2 (2012) ·Zbl 1243.17004 |
| [2] | Comes, J.; Heidersdorf, T., Thick ideals in Deligne’s category \(\text{Rep}_{\_}( O_\delta)\), J. Algebra, 480 (2017) ·Zbl 1423.18021 |
| [3] | Duflo, M.; Serganova, V., On associated variety for Lie superalgebras |
| [4] | Ehrig, M.; Stroppel, C., On the category of finite-dimensional representations of \(O S P(r | 2 n)\), (Krause, H.; etal., Representation Theory - Current Trends and Perspectives. Representation Theory - Current Trends and Perspectives, EMS Ser. Congr. Rep. (2017)), 109-170 ·Zbl 1425.17007 |
| [5] | Ehrig, M.; Stroppel, C., On the category of finite-dimensional representations of \(O S P(r | 2 n)\), Part II, available at |
| [6] | Ehrig, M.; Stroppel, C., Nazarov-Wenzl algebras, coideal subalgebras and categorified skew Howe duality, Adv. Math., 331 (2018) ·Zbl 1432.16022 |
| [7] | Entova-Aizenbud, I.; Serganova, V., Duflo-Serganova functor and superdimension formula for the periplectic Lie superalgebra (2019), to appear in Algebra Number Theory |
| [8] | Gorelik, M., Depths and cores in the light of DS-functors (2020) |
| [9] | Gorelik, M., Bipartite extension graphs and the DS functor (2020) |
| [10] | Gorelik, M.; Heidersdorf, T., Gruson-Serganova character formula and the Duflo-Serganova cohomology functor (2021) |
| [11] | Gruson, C.; Serganova, V., Cohomology of generalized supergrassmanians and character formulae for basic classical Lie superalgebras, Proc. Lond. Math. Soc. (3), 101, 852-892 (2010) ·Zbl 1216.17005 |
| [12] | Gruson, C.; Serganova, V., Bernstein-Gelfand-Gelfand reciprocity and indecomposable projective modules for classical algebraic supergroups, Mosc. Math. J., 13, 2, 281-313 (2013) ·Zbl 1345.17009 |
| [13] | Heidersdorf, T., On supergroups and their semisimplified representation categories, Algebr. Represent. Theory, 22, 937-959 (2019) ·Zbl 1465.17011 |
| [14] | Heidersdorf, T.; Weissauer, R., Cohomological tensor functors on representations of the general linear supergroup, Mem. Am. Math. Soc., 270, Article 1320 pp. (2021) ·Zbl 1475.17013 |
| [15] | Heidersdorf, T.; Weissauer, R., On classical tensor categories attached to the irreducible representations of the general linear supergroups \(G L(n | n) (2018)\) |
| [16] | Hoyt, C.; Reif, S., Grothendieck rings for Lie superalgebras and the Duflo-Serganova functor, Algebra Number Theory, 12, 9, 2167-2184 (2018) ·Zbl 1462.17014 |
| [17] | Serganova, V., On a superdimension of an irreducible representation of a basic classical Lie superalgebras, (Supersymmetry in Mathematics and Physics. Supersymmetry in Mathematics and Physics, Lecture Notes in Math., vol. 2027 (2011), Springer: Springer Heidelberg), 253-273 ·Zbl 1287.17020 |
| [18] | Serganova, V., Finite dimensional representations of algebraic supergroups, (Proceedings of the International Congress of Mathematicians ICM 2014, Seoul, Korea, August 13-21, 2014, vol. I: Plenary Lectures and Ceremonies (2014)) ·Zbl 1373.17023 |
| [19] | Weissauer, R., Model structures, categorical quotients and representations of super commutative Hopf algebras II, The case \(G L(m | n)\) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
