Serre functors for Lie algebras and superalgebras
Annales de l'Institut Fourier, Volume 62 (2012) no. 1, pp. 47-75.

We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category 𝒪 associated to a semi-simple complex finite dimensional Lie algebra. We further show that our realization carries over to classical Lie superalgebras in many cases. Along the way we prove that category 𝒪 and its parabolic generalizations for classical Lie superalgebras are categories with full projective functors. As an application we prove that in many cases the endomorphism algebra of the basic projective-injective module in (parabolic) category 𝒪 for classical Lie superalgebras is symmetric. As a special case we obtain that in these cases the algebras describing blocks of the category of finite dimensional modules are symmetric. We also compute the latter algebras for the superalgebra 𝔮(2).

Nous proposons une nouvelle réalisation du foncteur de Serre pour la catégorie 𝒪 de BGG associée à une algèbre de Lie semi-simple complexe de dimension finie, en utilisant les bimodules d’Harish-Chandra. De plus, nous démontrons que dans beaucoup de cas notre réalisation s’applique aux super algèbres de Lie classiques. Pour cela, nous prouvons que la catégorie 𝒪 et ses généralisations paraboliques pour les super-algèbres de Lie classiques sont des catégories avec foncteurs pleins projectifs. Comme application, nous montrons que, dans beaucoup de cas, l’algèbre d’endomorphismes du module projectif-injectif basique de la catégorie 𝒪 (parabolique) pour les super-algèbres de Lie est symétrique. En particulier, dans ce cas, les algèbres décrivant les blocs de la catégorie de modules de dimension finie sont symétriques. Nous calculons ces dernières algèbres pour la super algèbre de Lie 𝔮(2).

DOI: 10.5802/aif.2698
Classification: 17B10, 16S30, 18G05
Keywords: Lie superalgebra, module, Harish-Chandra bimodule, Serre functor, quiver, category $\mathcal{O}$
Mot clés : super algèbres de Lie, bimodules d’Harish-Chandra, foncteur de Serre, carquois, catégorie $\mathcal{O}$
Mazorchuk, Volodymyr 1; Miemietz, Vanessa 2

1 Uppsala University Department of Mathematics Box 480 751 06, Uppsala (Sweden)
2 University of East Anglia School of Mathematics Norwich NR4 7TJ (United Kingdom)
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Mazorchuk, Volodymyr; Miemietz, Vanessa. Serre functors for Lie algebras and superalgebras. Annales de l'Institut Fourier, Volume 62 (2012) no. 1, pp. 47-75. doi : 10.5802/aif.2698. https://aif.centre-mersenne.org/articles/10.5802/aif.2698/

[1] Beilinson, A.; Bezrukavnikov, R.; Mirković, I. Tilting exercises, Mosc. Math. J., Volume 4 (2004) no. 3, p. 547-557, 782 | MR

[2] Bernstein, J. N.; Gel’fand, S. I. Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras, Compositio Math., Volume 41 (1980) no. 2, pp. 245-285 | Numdam | MR | Zbl

[3] Boe, Brian D.; Kujawa, Jonathan R.; Nakano, Daniel K. Complexity and module varieties for classical Lie superalgebras (Preprint, http://arxiv.org/abs/0905.2403) | MR

[4] Bondal, A. I.; Kapranov, M. M. Representable functors, Serre functors, and reconstructions, Izv. Akad. Nauk SSSR Ser. Mat., Volume 53 (1989) no. 6, p. 1183-1205, 1337 | MR | Zbl

[5] Brundan, Jonathan Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra 𝔤𝔩(m|n), J. Amer. Math. Soc., Volume 16 (2003) no. 1, pp. 185-231 | DOI | MR

[6] Brundan, Jonathan Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra 𝔮(n), Adv. Math., Volume 182 (2004) no. 1, pp. 28-77 | DOI | MR

[7] Brundan, Jonathan Tilting modules for Lie superalgebras, Comm. Algebra, Volume 32 (2004) no. 6, pp. 2251-2268 | DOI | MR

[8] Brundan, Jonathan; Stroppel, Catharina Highest weight categories arising from Khovanov’s diagram algebra. IV. The general linear supergroup (to appear in JEMS)

[9] Cheng, Shun-Jen; Lam, Ngau; Wang, Weiqiang Super duality and irreducible characters of ortho-symplectic Lie superalgebras, Inv. Math., Volume 183 (2011) no. 1, pp. 189-244 | DOI | MR

[10] Frisk, Anders Typical blocks of the category 𝒪 for the queer Lie superalgebra, J. Algebra Appl., Volume 6 (2007) no. 5, pp. 731-778 | DOI | MR

[11] Frisk, Anders; Mazorchuk, Volodymyr Regular strongly typical blocks of 𝒪 𝔮 , Comm. Math. Phys., Volume 291 (2009) no. 2, pp. 533-542 | DOI | MR

[12] Gorelik, Maria On the ghost centre of Lie superalgebras, Ann. Inst. Fourier (Grenoble), Volume 50 (2000) no. 6, p. 1745-1764 (2001) | DOI | Numdam | MR

[13] Gorelik, Maria Annihilation theorem and separation theorem for basic classical Lie superalgebras, J. Amer. Math. Soc., Volume 15 (2002) no. 1, p. 113-165 (electronic) | DOI | MR

[14] Gorelik, Maria Strongly typical representations of the basic classical Lie superalgebras, J. Amer. Math. Soc., Volume 15 (2002) no. 1, p. 167-184 (electronic) | DOI | MR

[15] Gorelik, Maria Shapovalov determinants of Q-type Lie superalgebras, IMRP Int. Math. Res. Pap. (2006), pp. Art. ID 96895, 71 | MR

[16] Happel, Dieter Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, 119, Cambridge University Press, Cambridge, 1988 | MR | Zbl

[17] Humphreys, James E. Representations of semisimple Lie algebras in the BGG category 𝒪, Graduate Studies in Mathematics, 94, American Mathematical Society, Providence, RI, 2008 | MR

[18] Jantzen, Jens Carsten Einhüllende Algebren halbeinfacher Lie-Algebren, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 3, Springer-Verlag, Berlin, 1983 | MR | Zbl

[19] Kåhrström, Johan; Mazorchuk, Volodymyr A new approach to Kostant’s problem, Algebra Number Theory, Volume 4 (2010) no. 3, pp. 231-254 | DOI | MR

[20] Khomenko, Oleksandr Categories with projective functors, Proc. London Math. Soc. (3), Volume 90 (2005) no. 3, pp. 711-737 | DOI | MR

[21] Khomenko, Oleksandr; Mazorchuk, Volodymyr On Arkhipov’s and Enright’s functors, Math. Z., Volume 249 (2005) no. 2, pp. 357-386 | DOI | MR

[22] Mazorchuk, Volodymyr Classification of simple 𝔮 2 -supermodules, Tohoku Math. J. (2), Volume 62 (2010) no. 3, pp. 401-426 | DOI | MR

[23] Mazorchuk, Volodymyr Some homological properties of the category 𝒪. II, Represent. Theory, Volume 14 (2010), pp. 249-263 | DOI | MR

[24] Mazorchuk, Volodymyr; Ovsienko, Serge; Stroppel, Catharina Quadratic duals, Koszul dual functors, and applications, Trans. Amer. Math. Soc., Volume 361 (2009) no. 3, pp. 1129-1172 | DOI | MR

[25] Mazorchuk, Volodymyr; Stroppel, Catharina Categorification of (induced) cell modules and the rough structure of generalised Verma modules, Adv. Math., Volume 219 (2008) no. 4, pp. 1363-1426 | DOI | MR

[26] Mazorchuk, Volodymyr; Stroppel, Catharina Projective-injective modules, Serre functors and symmetric algebras, J. Reine Angew. Math., Volume 616 (2008), pp. 131-165 | DOI | MR

[27] Miličić, Dragan; Soergel, Wolfgang The composition series of modules induced from Whittaker modules, Comment. Math. Helv., Volume 72 (1997) no. 4, pp. 503-520 | DOI | MR | Zbl

[28] Musson, Ian M. A classification of primitive ideals in the enveloping algebra of a classical simple Lie superalgebra, Adv. Math., Volume 91 (1992) no. 2, pp. 252-268 | DOI | MR | Zbl

[29] Penkov, I.; Serganova, V. Characters of irreducible G-modules and cohomology of G/P for the Lie supergroup G=Q(N), J. Math. Sci. (New York), Volume 84 (1997) no. 5, pp. 1382-1412 (Algebraic geometry, 7) | DOI | MR | Zbl

[30] Penkov, Ivan; Serganova, Vera Characters of finite-dimensional irreducible 𝔮(n)-modules, Lett. Math. Phys., Volume 40 (1997) no. 2, pp. 147-158 | DOI | MR | Zbl

[31] Rocha-Caridi, Alvany Splitting criteria for 𝔤-modules induced from a parabolic and the Berňsteĭn-Gel fand-Gel fand resolution of a finite-dimensional, irreducible 𝔤-module, Trans. Amer. Math. Soc., Volume 262 (1980) no. 2, pp. 335-366 | MR | Zbl

[32] Scott, Leonard L. Simulating algebraic geometry with algebra. I. The algebraic theory of derived categories, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) (Proc. Sympos. Pure Math.), Volume 47, Amer. Math. Soc., Providence, RI, 1987, pp. 271-281 | MR | Zbl

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