Algebraic supergroups and Harish-Chandra pairs over a commutative ring
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- by Akira Masuoka and Taiki Shibata PDF
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Abstract:
We prove a category equivalence between algebraic supergroups and Harish-Chandra pairs over a commutative ring which is $2$-torsion free. The result is applied to reconstruct the Chevalley $\mathbb {Z}$-supergroups constructed by Fioresi and Gavarini (2012) and by Gavarini (2014). For a wide class of algebraic supergroups we describe their representations by using their super-hyperalgebras.References
- George M. Bergman, The diamond lemma for ring theory, Adv. in Math. 29 (1978), no. 2, 178–218. MR 506890, DOI 10.1016/0001-8708(78)90010-5
- Jonathan Brundan and Alexander Kleshchev, Modular representations of the supergroup $Q(n)$. I, J. Algebra 260 (2003), no. 1, 64–98. Special issue celebrating the 80th birthday of Robert Steinberg. MR 1973576, DOI 10.1016/S0021-8693(02)00620-8
- Jonathan Brundan and Jonathan Kujawa, A new proof of the Mullineux conjecture, J. Algebraic Combin. 18 (2003), no. 1, 13–39. MR 2002217, DOI 10.1023/A:1025113308552
- Claudio Carmeli, Lauren Caston, and Rita Fioresi, Mathematical foundations of supersymmetry, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2011. MR 2840967, DOI 10.4171/097
- Claudio Carmeli and Rita Fioresi, Superdistributions, analytic and algebraic super Harish-Chandra pairs, Pacific J. Math. 263 (2013), no. 1, 29–51. MR 3069075, DOI 10.2140/pjm.2013.263.29
- Michel Demazure and Pierre Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeurs, Paris; North-Holland Publishing Co., Amsterdam, 1970 (French). Avec un appendice Corps de classes local par Michiel Hazewinkel. MR 0302656
- R. Fioresi and F. Gavarini, On the construction of Chevalley supergroups, Supersymmetry in mathematics and physics, Lecture Notes in Math., vol. 2027, Springer, Heidelberg, 2011, pp. 101–123. MR 2906339, DOI 10.1007/978-3-642-21744-9_{5}
- R. Fioresi and F. Gavarini, Chevalley supergroups, Mem. Amer. Math. Soc. 215 (2012), no. 1014, vi+64. MR 2918543, DOI 10.1090/S0065-9266-2011-00633-7
- F. Gavarini, Chevalley supergroups of type $D(2,1;a)$, Proc. Edinb. Math. Soc. (2) 57 (2014), no. 2, 465–491. MR 3200319, DOI 10.1017/S0013091513000503
- Fabio Gavarini, Algebraic supergroups of Cartan type, Forum Math. 26 (2014), no. 5, 1473–1564. MR 3334037, DOI 10.1515/forum-2011-0144
- Fabio Gavarini, Global splittings and super Harish-Chandra pairs for affine supergroups, Trans. Amer. Math. Soc. 368 (2016), no. 6, 3973–4026. MR 3453363, DOI 10.1090/tran/6456
- A. N. Grishkov and A. N. Zubkov, Solvable, reductive and quasireductive supergroups, J. Algebra 452 (2016), 448–473. MR 3461076, DOI 10.1016/j.jalgebra.2015.11.013
- P. J. Higgins, Baer invariants and the Birkhoff-Witt theorem, J. Algebra 11 (1969), 469–482. MR 238913, DOI 10.1016/0021-8693(69)90086-6
- Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057
- V. G. Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8–96. MR 486011, DOI 10.1016/0001-8708(77)90017-2
- Bertram Kostant, Groups over $Z$, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 90–98. MR 0207713
- Bertram Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential geometrical methods in mathematical physics (Proc. Sympos., Univ. Bonn, Bonn, 1975) Lecture Notes in Math., Vol. 570, Springer, Berlin, 1977, pp. 177–306. MR 0580292
- J.-L. Koszul, Graded manifolds and graded Lie algebras, Proceedings of the international meeting on geometry and physics (Florence, 1982) Pitagora, Bologna, 1983, pp. 71–84. MR 760837
- Akira Masuoka, The fundamental correspondences in super affine groups and super formal groups, J. Pure Appl. Algebra 202 (2005), no. 1-3, 284–312. MR 2163412, DOI 10.1016/j.jpaa.2005.02.010
- Akira Masuoka, Harish-Chandra pairs for algebraic affine supergroup schemes over an arbitrary field, Transform. Groups 17 (2012), no. 4, 1085–1121. MR 3000482, DOI 10.1007/s00031-012-9203-8
- A. Masuoka, Hopf algebraic techniques applied to super algebraic groups, Proceedings of Algebra Symposium (Hiroshima, 2013), pp. 48–66, Math. Soc. Japan, 2013; available at arXiv:1311.1261.
- Akira Masuoka and Alexandr N. Zubkov, Quotient sheaves of algebraic supergroups are superschemes, J. Algebra 348 (2011), 135–170. MR 2852235, DOI 10.1016/j.jalgebra.2011.08.038
- Susan Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993. MR 1243637, DOI 10.1090/cbms/082
- Bin Shu and Weiqiang Wang, Modular representations of the ortho-symplectic supergroups, Proc. Lond. Math. Soc. (3) 96 (2008), no. 1, 251–271. MR 2392322, DOI 10.1112/plms/pdm040
- Moss E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR 0252485
- Mitsuhiro Takeuchi, On coverings and hyperalgebras of affine algebraic groups, Trans. Amer. Math. Soc. 211 (1975), 249–275. MR 429928, DOI 10.1090/S0002-9947-1975-0429928-3
- M. Takeuchi, Hyperalgebraic construction of Chevalley group schemes (in Japanese), RIMS Kokyuroku 473 (1982), 57–70.
- E. G. Vishnyakova, On complex Lie supergroups and split homogeneous supermanifolds, Transform. Groups 16 (2011), no. 1, 265–285. MR 2785503, DOI 10.1007/s00031-010-9114-5
Additional Information
- Akira Masuoka
- Affiliation: Institute of Mathematics, University of Tsukuba, Ibaraki 305-8571, Japan
- MR Author ID: 261525
- Email: akira@math.tsukuba.ac.jp
- Taiki Shibata
- Affiliation: Graduate School of Pure and Applied Sciences, University of Tsukuba, Ibaraki 305-8571, Japan
- Address at time of publication: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
- MR Author ID: 977562
- ORCID: 0000-0002-9031-9677
- Email: shibata@ualberta.ca
- Received by editor(s): May 30, 2013
- Received by editor(s) in revised form: March 22, 2015, and May 9, 2015
- Published electronically: September 27, 2016
- Additional Notes: The first author was supported by JSPS Grant-in-Aid for Scientific Research (C) 23540039
The second author was supported by Grant-in-Aid for JSPS Fellows 26E2022 - © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 3443-3481
- MSC (2010): Primary 14M30, 16T05, 16W55
- DOI: https://doi.org/10.1090/tran/6751
- MathSciNet review: 3605977