Serre functors for Lie algebras and superalgebras
Annales de l'Institut Fourier, Volume 62 (2012) no. 1, p. 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 : https://doi.org/10.5802/aif.2698
Classification:  17B10,  16S30,  18G05
Keywords: Lie superalgebra, module, Harish-Chandra bimodule, Serre functor, quiver, category 𝒪
@article{AIF_2012__62_1_47_0,
     author = {Mazorchuk, Volodymyr and Miemietz, Vanessa},
     title = {Serre functors for Lie algebras and superalgebras},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {1},
     year = {2012},
     pages = {47-75},
     doi = {10.5802/aif.2698},
     zbl = {pre06064511},
     mrnumber = {2986264},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2012__62_1_47_0}
}
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/item/AIF_2012__62_1_47_0/

[1] Beilinson, A.; Bezrukavnikov, R.; Mirković, I. Tilting exercises, Mosc. Math. J., Tome 4 (2004) no. 3, p. 547-557, 782 | MR 2119139

[2] Bernstein, J. N.; Gel’Fand, S. I. Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras, Compositio Math., Tome 41 (1980) no. 2, pp. 245-285 | Numdam | MR 581584 | Zbl 0445.17006

[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 2764876

[4] Bondal, A. I.; Kapranov, M. M. Representable functors, Serre functors, and reconstructions, Izv. Akad. Nauk SSSR Ser. Mat., Tome 53 (1989) no. 6, p. 1183-1205, 1337 | MR 1039961 | Zbl 0703.14011

[5] Brundan, Jonathan Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra 𝔤𝔩(m|n), J. Amer. Math. Soc., Tome 16 (2003) no. 1, pp. 185-231 | Article | MR 1937204

[6] Brundan, Jonathan Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra 𝔮(n), Adv. Math., Tome 182 (2004) no. 1, pp. 28-77 | Article | MR 2028496

[7] Brundan, Jonathan Tilting modules for Lie superalgebras, Comm. Algebra, Tome 32 (2004) no. 6, pp. 2251-2268 | Article | MR 2100468

[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., Tome 183 (2011) no. 1, pp. 189-244 | Article | MR 2755062

[10] Frisk, Anders Typical blocks of the category 𝒪 for the queer Lie superalgebra, J. Algebra Appl., Tome 6 (2007) no. 5, pp. 731-778 | Article | MR 2355618

[11] Frisk, Anders; Mazorchuk, Volodymyr Regular strongly typical blocks of 𝒪 𝔮 , Comm. Math. Phys., Tome 291 (2009) no. 2, pp. 533-542 | Article | MR 2530171

[12] Gorelik, Maria On the ghost centre of Lie superalgebras, Ann. Inst. Fourier (Grenoble), Tome 50 (2000) no. 6, p. 1745-1764 (2001) | Article | Numdam | MR 1817382

[13] Gorelik, Maria Annihilation theorem and separation theorem for basic classical Lie superalgebras, J. Amer. Math. Soc., Tome 15 (2002) no. 1, p. 113-165 (electronic) | Article | MR 1862799

[14] Gorelik, Maria Strongly typical representations of the basic classical Lie superalgebras, J. Amer. Math. Soc., Tome 15 (2002) no. 1, p. 167-184 (electronic) | Article | MR 1862800

[15] Gorelik, Maria Shapovalov determinants of Q-type Lie superalgebras, IMRP Int. Math. Res. Pap. (2006), pp. Art. ID 96895, 71 | MR 2282179

[16] Happel, Dieter Triangulated categories in the representation theory of finite-dimensional algebras, Cambridge University Press, Cambridge, London Mathematical Society Lecture Note Series, Tome 119 (1988) | MR 935124 | Zbl 0635.16017

[17] Humphreys, James E. Representations of semisimple Lie algebras in the BGG category 𝒪, American Mathematical Society, Providence, RI, Graduate Studies in Mathematics, Tome 94 (2008) | MR 2428237

[18] Jantzen, Jens Carsten Einhüllende Algebren halbeinfacher Lie-Algebren, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Tome 3 (1983) | MR 721170 | Zbl 0541.17001

[19] Kåhrström, Johan; Mazorchuk, Volodymyr A new approach to Kostant’s problem, Algebra Number Theory, Tome 4 (2010) no. 3, pp. 231-254 | Article | MR 2602666

[20] Khomenko, Oleksandr Categories with projective functors, Proc. London Math. Soc. (3), Tome 90 (2005) no. 3, pp. 711-737 | Article | MR 2137828

[21] Khomenko, Oleksandr; Mazorchuk, Volodymyr On Arkhipov’s and Enright’s functors, Math. Z., Tome 249 (2005) no. 2, pp. 357-386 | Article | MR 2115448

[22] Mazorchuk, Volodymyr Classification of simple 𝔮 2 -supermodules, Tohoku Math. J. (2), Tome 62 (2010) no. 3, pp. 401-426 | Article | MR 2742017

[23] Mazorchuk, Volodymyr Some homological properties of the category 𝒪. II, Represent. Theory, Tome 14 (2010), pp. 249-263 | Article | MR 2602033

[24] Mazorchuk, Volodymyr; Ovsienko, Serge; Stroppel, Catharina Quadratic duals, Koszul dual functors, and applications, Trans. Amer. Math. Soc., Tome 361 (2009) no. 3, pp. 1129-1172 | Article | MR 2457393

[25] Mazorchuk, Volodymyr; Stroppel, Catharina Categorification of (induced) cell modules and the rough structure of generalised Verma modules, Adv. Math., Tome 219 (2008) no. 4, pp. 1363-1426 | Article | MR 2450613

[26] Mazorchuk, Volodymyr; Stroppel, Catharina Projective-injective modules, Serre functors and symmetric algebras, J. Reine Angew. Math., Tome 616 (2008), pp. 131-165 | Article | MR 2369489

[27] Miličić, Dragan; Soergel, Wolfgang The composition series of modules induced from Whittaker modules, Comment. Math. Helv., Tome 72 (1997) no. 4, pp. 503-520 | Article | MR 1600134 | Zbl 0956.17004

[28] Musson, Ian M. A classification of primitive ideals in the enveloping algebra of a classical simple Lie superalgebra, Adv. Math., Tome 91 (1992) no. 2, pp. 252-268 | Article | MR 1149625 | Zbl 0799.17008

[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), Tome 84 (1997) no. 5, pp. 1382-1412 (Algebraic geometry, 7) | Article | MR 1465520 | Zbl 0920.17003

[30] Penkov, Ivan; Serganova, Vera Characters of finite-dimensional irreducible 𝔮(n)-modules, Lett. Math. Phys., Tome 40 (1997) no. 2, pp. 147-158 | Article | MR 1463616 | Zbl 0892.17006

[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., Tome 262 (1980) no. 2, pp. 335-366 | MR 586721 | Zbl 0449.17008

[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), Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 47 (1987), pp. 271-281 | MR 933417 | Zbl 0659.20038