Lie supergroup actions on supermanifolds
Authors:
Charles P. Boyer and O. A. SánchezValenzuela
Journal:
Trans. Amer. Math. Soc. 323 (1991), 151175
MSC:
Primary 58A50; Secondary 17A70, 22E30, 22E60
MathSciNet review:
998351
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Lie supergroups are here understood as group objects in the category of supermanifolds (as in [, , and ]). Actions of Lie supergroups in supermanifolds are defined by means of diagrams of supermanifold morphisms. Examples of such actions are given. Among them emerge the linear actions discussed in [, , and ] and the natural actions on the Grassmannian supermanifolds studied in [ and ]. The nature of the isotropy subsupergroup associated to an action is fully elucidated; it is exhibited as an embedded subsupergroup in the spirit of the theory of smooth manifolds and Lie groups and with no need for the LieHopf algebraic approach of Kostant in []. The notion of orbit is also discussed. Explicit calculations of isotropy subsupergroups are included. Also, an alternative proof of the fact that the structural sheaf of a Lie supergroup is isomorphic to the sheaf of sections of a trivial exterior algebra bundle is given, based on the triviality of its supertangent bundle.
 [1]
Marjorie
Batchelor, The structure of
supermanifolds, Trans. Amer. Math. Soc. 253 (1979), 329–338.
MR 536951
(80h:58002), http://dx.doi.org/10.1090/S00029947197905369510
 [2]
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 (89b:58006)
 [3]
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
(58 #28326)
 [4]
D.
A. Leĭtes, Introduction to the theory of
supermanifolds, Uspekhi Mat. Nauk 35 (1980),
no. 1(211), 3–57, 255 (Russian). MR 565567
(81j:58003)
 [5]
, Seminar on supermanifolds, no. 31; Report from the Department of Mathematics, Univ. of Stockholm, Sweden, 1987.
 [6]
Yu.
I. Manin, Holomorphic supergeometry and YangMills
superfields, Current problems in mathematics, Vol. 24, Itogi Nauki i
Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform.,
Moscow, 1984, pp. 3–80 (Russian). MR 760997
(86e:32038)
 [7]
Yuri
I. Manin, Gauge field theory and complex geometry, Grundlehren
der Mathematischen Wissenschaften [Fundamental Principles of Mathematical
Sciences], vol. 289, SpringerVerlag, Berlin, 1988. Translated from
the Russian by N. Koblitz and J. R. King. MR 954833
(89d:32001)
 [8]
, Grassmannians and flags in supergeometry, Amer. Math. Soc. Transl. (2) 137 (1987), 8799.
 [9]
Michael
Artin and John
Tate (eds.), Arithmetic and geometry. Vol. II, Progress in
Mathematics, vol. 36, Birkhäuser, Boston, Mass., 1983. Geometry;
Papers dedicated to I. R. Shafarevich on the occasion of his sixtieth
birthday. MR
717602 (84j:14005b)
 [10]
Mitchell
J. Rothstein, The axioms of supermanifolds and a new
structure arising from them, Trans. Amer. Math.
Soc. 297 (1986), no. 1, 159–180. MR 849473
(87m:58015), http://dx.doi.org/10.1090/S00029947198608494738
 [11]
O. A. Sanchez Valenzuela, On supergeometric structures, Harvard thesis, May 1986, and On supervector bundles, IIMASUNAM preprint 457, Mexico, 1986.
 [12]
Oscar
Adolfo SánchezValenzuela, Linear supergroup actions. I. On the
defining properties, Trans. Amer. Math.
Soc. 307 (1988), no. 2, 569–595. MR 940218
(90a:58010), http://dx.doi.org/10.1090/S00029947198809402181
 [13]
Oscar
Adolfo SánchezValenzuela, Remarks on Grassmannian
supermanifolds, Trans. Amer. Math. Soc.
307 (1988), no. 2,
597–614. MR
940219 (89k:58027), http://dx.doi.org/10.1090/S00029947198809402193
 [14]
, Matrix computations in linear superalgebra, Linear Algebra Appl. (to appear).
 [15]
Thomas
Schmitt, Super differential geometry, Report MATH,
vol. 84, Akademie der Wissenschaften der DDR, Institut für
Mathematik, Berlin, 1984. With German and Russian summaries. MR 786297
(86m:58003)
 [16]
B.
R. Tennison, Sheaf theory, Cambridge University Press,
Cambridge, EnglandNew YorkMelbourne, 1975. London Mathematical Society
Lecture Note Series, No. 20. MR 0404390
(53 #8192)
 [1]
 M. Batchelor, The structure of supermanifolds, Trans. Amer. Math. Soc. 253 (1980), 329338. MR 536951 (80h:58002)
 [2]
 F. A. Berezin, Introduction to superanalysis (A. A. Kirillov, ed.), Kluwer Academic Publishers, 1987. MR 914369 (89b:58006)
 [3]
 B. Kostant, Graded manifolds, graded Lie theory and prequantization, Lecture Notes in Math., vol. 570, SpringerVerlag, Berlin, 1977, pp. 177306. MR 0580292 (58:28326)
 [4]
 D. A. Leites, Introduction to the theory of supermanifolds, Russian Math. Surveys 35 (1980), 164. MR 565567 (81j:58003)
 [5]
 , Seminar on supermanifolds, no. 31; Report from the Department of Mathematics, Univ. of Stockholm, Sweden, 1987.
 [6]
 Yu. I. Manin, Holomorphic supergeometry and YangMills superfields, preprint. MR 760997 (86e:32038)
 [7]
 , Gauge field theory and complex geometry, Nauka, Moscow, 1984; English transl., SpringerVerlag, New York, 1988. MR 954833 (89d:32001)
 [8]
 , Grassmannians and flags in supergeometry, Amer. Math. Soc. Transl. (2) 137 (1987), 8799.
 [9]
 , Flag superspaces and supersymmetric YangMills equations, Arithmetic and Geometry: Papers Dedicated to I. R. Shafarevich on the Occasion of his 60th Birthday, vol. II (M. Artin and J. Tate, eds.), Birkhäuser, Boston, Mass., 1983. MR 717602 (84j:14005b)
 [10]
 M. Rothstein, The axioms of supermanifolds and a new structure arising from them, Trans. Amer. Math. Soc. 297 (1986), 159180. MR 849473 (87m:58015)
 [11]
 O. A. Sanchez Valenzuela, On supergeometric structures, Harvard thesis, May 1986, and On supervector bundles, IIMASUNAM preprint 457, Mexico, 1986.
 [12]
 , Linear supergroup actions. I: On the defining properties, Trans. Amer. Math. Soc. 307 (1988), 569595. MR 940218 (90a:58010)
 [13]
 , On Grassmannian supermanifolds, Trans. Amer. Math. Soc. 307 (1988), 597614. MR 940219 (89k:58027)
 [14]
 , Matrix computations in linear superalgebra, Linear Algebra Appl. (to appear).
 [15]
 T. Schmitt, Super differential geometry, Akademie der Wissenschaften der DDR, Institut für Mathematik, report RMATH05/84, Berlin, 1984. MR 786297 (86m:58003)
 [16]
 O. Tennison, Sheaf theory, London Math. Soc. Lecture Notes Ser., no. 20, Cambridge Univ. Press, Cambridge, 1975. MR 0404390 (53:8192)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
58A50,
17A70,
22E30,
22E60
Retrieve articles in all journals
with MSC:
58A50,
17A70,
22E30,
22E60
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199109983514
PII:
S 00029947(1991)09983514
Article copyright:
© Copyright 1991
American Mathematical Society
