Skip to Main Content

AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution

Chevalley supergroups

About this Title

R. Fioresi, Dipartimento di Matematica, Università di Bologna, piazza di Porta San Donato, 5 — I-40127 Bologna, Italy and F. Gavarini, Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della ricerca scientifica 1 — I-00133 Roma, Italy

Publication: Memoirs of the American Mathematical Society
Publication Year: 2012; Volume 215, Number 1014
ISBNs: 978-0-8218-5300-9 (print); 978-0-8218-8521-5 (online)
Published electronically: May 3, 2011
Keywords: Algebraic supergroups
MSC: Primary 14M30, 14A22; Secondary 58A50, 17B50

View full volume PDF

View other years and numbers:

Table of Contents


  • 1. Introduction
  • 2. Preliminaries
  • 3. Chevalley bases and Chevalley algebras
  • 4. Kostant superalgebras
  • 5. Chevalley supergroups
  • 6. The cases $A(1,1) \,$, $P(3)$ and $Q(n)$
  • A. Sheafification


In the framework of algebraic supergeometry, we give a construction of the scheme-theoretic supergeometric analogue of split reductive algebraic group- schemes, namely affine algebraic supergroups associated to simple Lie superalgebras of classical type. In particular, all Lie superalgebras of both basic and strange types are considered. This provides a unified approach to most of the algebraic supergroups considered so far in the literature, and an effective method to construct new ones.

Our method follows the pattern of a suitable scheme-theoretic revisitation of Chevalley’s construction of semisimple algebraic groups, adapted to the reductive case. As an intermediate step, we prove an existence theorem for Chevalley bases of simple classical Lie superalgebras and a PBW-like theorem for their associated Kostant superalgebras.

References [Enhancements On Off] (What's this?)

  • M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
  • Luigi Balduzzi, Claudio Carmeli, and Gianni Cassinelli, Super $G$-spaces, Symmetry in mathematics and physics, Contemp. Math., vol. 490, Amer. Math. Soc., Providence, RI, 2009, pp. 159–176. MR 2555976, DOI 10.1090/conm/490/09594
  • Felix Alexandrovich Berezin, Introduction to superanalysis, Mathematical Physics and Applied Mathematics, vol. 9, D. Reidel Publishing Co., Dordrecht, 1987. Edited and with a foreword by A. A. Kirillov; With an appendix by V. I. Ogievetsky; Translated from the Russian by J. Niederle and R. Kotecký; Translation edited by Dimitri Leĭtes. MR 914369
  • 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
  • Armand Borel, Properties and linear representations of Chevalley groups, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Lecture Notes in Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 1–55. MR 0258838
  • C. Carmeli, L. Caston, R. Fioresi, with an appendix by I. Dimitrov, Mathematical Foundation of Supersymmetry, EMS Ser. Lect. Math., European Math. Soc., Zurich, 2011.
  • Pierre Deligne and John W. Morgan, Notes on supersymmetry (following Joseph Bernstein), Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997) Amer. Math. Soc., Providence, RI, 1999, pp. 41–97. MR 1701597
  • Michel Demazure and Pierre Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeur, Paris; North-Holland Publishing Co., Amsterdam, 1970 (French). Avec un appendice Corps de classes local par Michiel Hazewinkel. MR 0302656
  • David Eisenbud and Joe Harris, The geometry of schemes, Graduate Texts in Mathematics, vol. 197, Springer-Verlag, New York, 2000. MR 1730819
  • M. Artin, J. E. Bertin, M. Demazure, P. Gabriel, A. Grothendieck, M. Raynaud, and J.-P. Serre, Schémas en groupes. Fasc. 3: Exposés 8 à 11, Institut des Hautes Études Scientifiques, Paris, 1964 (French). Troisième édition; Séminaire de Géométrie Algébrique de l’Institut des Hautes Études Scientifiques, 1963/64, dirigé par Michel Demazure et Alexander Grothendieck. MR 0207705
  • Rita Fioresi, Smoothness of algebraic supervarieties and supergroups, Pacific J. Math. 234 (2008), no. 2, 295–310. MR 2373450, DOI 10.2140/pjm.2008.234.295
  • L. Frappat, A. Sciarrino, and P. Sorba, Dictionary on Lie algebras and superalgebras, Academic Press, Inc., San Diego, CA, 2000. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 1773773
  • F. Gavarini, Chevalley Supergroups of type $D(2,1;a)$, preprint arXiv:1006.0464 [math.RA] (2010).
  • Alexander Grothendieck, Technique de descente et théorèmes d’existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats, Séminaire Bourbaki, Vol. 5, Soc. Math. France, Paris, 1995, pp. Exp. No. 190, 299–327 (French). MR 1603475
  • Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
  • James E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York-Berlin, 1972. Graduate Texts in Mathematics, Vol. 9. MR 0323842
  • Kenji Iohara and Yoshiyuki Koga, Central extensions of Lie superalgebras, Comment. Math. Helv. 76 (2001), no. 1, 110–154. MR 1819663, DOI 10.1007/s000140050152
  • Jens Carsten Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1359532
  • 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, Graded manifolds, graded Lie theory, and prequantization, Differential geometrical methods in mathematical physics (Proc. Sympos., Univ. Bonn, Bonn, 1975) Springer, Berlin, 1977, pp. 177–306. Lecture Notes in Math., Vol. 570. 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
  • Serge Lang, Algebra, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR 1878556
  • Yuri I. Manin, Gauge field theory and complex geometry, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 289, Springer-Verlag, Berlin, 1997. Translated from the 1984 Russian original by N. Koblitz and J. R. King; With an appendix by Sergei Merkulov. MR 1632008
  • 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
  • Alice Rogers, Supermanifolds, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. Theory and applications. MR 2320438
  • Manfred Scheunert, The theory of Lie superalgebras, Lecture Notes in Mathematics, vol. 716, Springer, Berlin, 1979. An introduction. MR 537441
  • Vera Serganova, On generalizations of root systems, Comm. Algebra 24 (1996), no. 13, 4281–4299. MR 1414584, DOI 10.1080/00927879608825814
  • Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR 0466335
  • 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
  • E. G. Vishnyakova, On complex Lie supergroups and homogeneous split supermanifolds, preprint arXiv:0908.1164v1 [math.DG] (2009).
  • Angelo Vistoli, Grothendieck topologies, fibered categories and descent theory, Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Amer. Math. Soc., Providence, RI, 2005, pp. 1–104. MR 2223406, DOI 10.1007/s00222-005-0429-0
  • V. S. Varadarajan, Supersymmetry for mathematicians: an introduction, Courant Lecture Notes in Mathematics, vol. 11, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2004. MR 2069561