On the construction of Chevalley supergroups. (English) Zbl 1276.14075
Ferrara, Sergio (ed.) et al., Supersymmetry in mathematics and physics. UCLA Los Angeles, USA 2010. Papers based on the presentations at the workshop, Februar 2010. Berlin: Springer (ISBN 978-3-642-21743-2/pbk; 978-3-642-21744-9/ebook). Lecture Notes in Mathematics 2027, 101-123 (2011).
Summary: We give a description of the construction of Chevalley supergroups, providing some explanatory examples. We avoid the discussion of the \(A(1,1)\), \(P(3)\) and \(Q(n)\) cases, for which our construction holds, but the exposition becomes more complicated. We shall not in general provide complete proofs for our statements, instead we will make an effort to convey the key ideas underlying our construction. A fully detailed account of our work appeared in [Chevalley supergroups, Mem. Am. Math. Soc. 1014, 64 p. (2012; Zbl 1239.14045]).
For the entire collection see [Zbl 1223.81015].
For the entire collection see [Zbl 1223.81015].
MSC:
| 14M30 | Supervarieties |
| 14A22 | Noncommutative algebraic geometry |
| 17B50 | Modular Lie (super)algebras |
| 58A50 | Supermanifolds and graded manifolds |
Citations:
Zbl 1239.14045References:
| [1] | Atiyah, MF; MacDonald, IG, Introduction to Commutative Algebra (1969), London: Addison-Wesley, London · Zbl 0175.03601 |
| [2] | Brundan, J.; Kleshchev, A., Modular representations of the supergroup Q(n), I. J. Algebra, 206, 64-98 (2003) · Zbl 1027.17004 · doi:10.1016/S0021-8693(02)00620-8 |
| [3] | Brundan, J.; Kujava, J., A new proof of the mullineux conjecture, J. Algebr. Combinator., 18, 13-39 (2003) · Zbl 1043.20006 · doi:10.1023/A:1025113308552 |
| [4] | A. Borel, in Properties and Linear Representations of Chevalley Groups, ed. by A. Borel et al. Seminar on Algebraic Groups and Related Finite Groups, Lecture Notes in Mathematics, vol. 131 (Springer, Berlin, 1970), pp. 1-55 · Zbl 0192.36201 |
| [5] | C. Carmeli, L. Caston, R. Fioresi, Mathematical Foundation of Supersymmetry, with an appendix with I. Dimitrov. EMS Ser. Lect. Math. (EMS, Zurich, 2011) · Zbl 1226.58003 |
| [6] | P. Deligne, J. Morgan, in Notes on Supersymmetry (following J. Bernstein). Quantum Fields and Strings. A Course for Mathematicians, vol. 1 (AMS, RI, 1999) |
| [7] | Demazure, M.; Gabriel, P., Groupes Algébriques, Tome 1, Mason&Cie éditeur (1970), Amsterdam: North-Holland, Amsterdam · Zbl 0203.23401 |
| [8] | M. Demazure, A. Grothendieck, Schémas en groupes, III, Séminaire de Géométrie Algébrique du Bois Marie, vol. 3 (1964) · Zbl 0212.52804 |
| [9] | D. Eisenbud, J. Harris, The Geometry of Schemes. Graduate Texts in Mathematics, vol. 197 (Springer, Heidelberg, 2000) · Zbl 0960.14002 |
| [10] | R. Fioresi, F. Gavarini, Chevalley Supergroups, preprint arXiv:0808.0785 Memoirs of the AMS (2008) (to be published) |
| [11] | Frappat, L.; Sorba, P.; Sciarrino, A., Dictionary on Lie Algebras and Superalgebras (2000), CA: Academic, CA · Zbl 0965.17001 |
| [12] | Hartshorne, Robin, Algebraic Geometry (1977), New York, NY: Springer New York, New York, NY · Zbl 0367.14001 |
| [13] | Humphreys, James E., Introduction to Lie Algebras and Representation Theory (1972), New York, NY: Springer New York, New York, NY · Zbl 0254.17004 |
| [14] | Iohara, K.; Koga, Y., Central extensions of Lie Superalgebras, Comment. Math. Helv., 76, 110-154 (2001) · Zbl 1036.17004 · doi:10.1007/s000140050152 |
| [15] | J.C. Jantzen, Lectures on Quantum Groups. Grad. Stud. Math., vol. 6 (AMS, RI, 1996) · Zbl 0842.17012 |
| [16] | Kac, VG, Lie superalgebras. Adv. Math., 26, 8-26 (1977) · Zbl 0366.17012 |
| [17] | J.-L. Koszul, Graded manifolds and graded Lie algebras, in Proceedings of the International Meeting on Geometry and Physics, Florence, 1982 (Pitagora, Bologna, 1982), pp. 71-84 · Zbl 0548.22012 |
| [18] | Manin, Y., Gauge Field Theory and Complex Geometry (1988), Berlin: Springer, Berlin · Zbl 0641.53001 |
| [19] | Masuoka, A., The fundamental correspondences in super affine groups and super formal groups, J. Pure Appl. Algebra, 202, 284-312 (2005) · Zbl 1078.16045 · doi:10.1016/j.jpaa.2005.02.010 |
| [20] | Scheunert, M., The Theory of Lie Superalgebras, Lecture Notes in Mathematics (1979), Heidelberg: Springer, Heidelberg · Zbl 0407.17001 |
| [21] | Serganova, V., On generalizations of root systems, Comm. Algebra, 24, 4281-4299 (1996) · Zbl 0902.17002 · doi:10.1080/00927879608825814 |
| [22] | Shu, B.; Wang, W., Modular representations of the ortho-symplectic supergroups, Proc. Lond. Math. Soc. (3), 96, 1, 251-271 (2008) · Zbl 1219.20031 |
| [23] | Steinberg, R., Lectures on Chevalley Groups (1968), New Haven: Yale University, New Haven · Zbl 1196.22001 |
| [24] | V.S. Varadarajan, Supersymmetry for Mathematicians: An Introduction, Courant Lecture Notes vol. 1 (AMS, RI, 2004) · Zbl 1142.58009 |
| [25] | A. Vistoli, in Grothendieck Topologies, Fibered Categories and Descent Theory. Fundamental Algebraic Geometry, Math. Surveys Monogr., vol. 123 (AMS, RI, 2005), pp. 1-104 |
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.
