Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


The structure of supermanifolds
HTML articles powered by AMS MathViewer

by Marjorie Batchelor PDF
Trans. Amer. Math. Soc. 253 (1979), 329-338 Request permission


The increasing recognition of Lie superalgebras and their importance in physics inspired a search to find an object, a “supermanifold", which would realize the geometry implicit in Lie superalgebras. This paper analyzes the structure of supermanifolds as defined by B. Kostant. The result is the following structure theorem. The Main Theorem. If E is a real vector bundle over the smooth manifold X, let $\Lambda E$ be the associated exterior bundle and let $\Gamma (\Lambda E)$ be the sheaf of sections of $\Lambda E$. Then every supermanifold over X is isomorphic to $\Gamma (\Lambda E)$ for some vector bundle E over X. Although the vector bundle E is not unique but is determined only up to isomorphism, and the isomorphism guaranteed is not canonical, the existence of the isomorphism provides a base for a better understanding of geometry in the graded setting.
  • F. A. Berezin, The method of second quantization, Pure and Applied Physics, Vol. 24, Academic Press, New York-London, 1966. Translated from the Russian by Nobumichi Mugibayashi and Alan Jeffrey. MR 0208930
  • F. A. Berezin and G. I. Kac, Lie groups with commuting and anticommuting parameters, Mat. Sb. (N.S.) 82 (124) (1970), 343–359 (Russian). MR 0265520
  • F. Berezin and D. Leites, Supervarieties, Soviet Math. Dokl. 16 (1975), 1218-1222.
  • L. Corwin, Y. Ne’eman, and S. Sternberg, Graded Lie algebras in mathematics and physics (Bose-Fermi symmetry), Rev. Modern Phys. 47 (1975), 573–603. MR 0438925, DOI 10.1103/RevModPhys.47.573
  • J. Dell and L. Smolin, A differential-geometric formulation of supersymmetry, Harvard Preprint (in preparation).
  • Friedrich Hirzebruch, Topological methods in algebraic geometry, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Translated from the German and Appendix One by R. L. E. Schwarzenberger; With a preface to the third English edition by the author and Schwarzenberger; Appendix Two by A. Borel; Reprint of the 1978 edition. MR 1335917
  • 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) Lecture Notes in Math., Vol. 570, Springer, Berlin, 1977, pp. 177–306. MR 0580292
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 58A05, 83E99
  • Retrieve articles in all journals with MSC: 58A05, 83E99
Additional Information
  • © Copyright 1979 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 253 (1979), 329-338
  • MSC: Primary 58A05; Secondary 83E99
  • DOI:
  • MathSciNet review: 536951