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Transactions of the American Mathematical Society

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Two approaches to supermanifolds

Author: Marjorie Batchelor
Journal: Trans. Amer. Math. Soc. 258 (1980), 257-270
MSC: Primary 58A05; Secondary 81C99
MathSciNet review: 554332
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Abstract: The problem of supplying an analogue of a manifold whose sheaf of functions contains anticommuting elements has been approached in two ways. Either one extends the sheaf of functions formally, as in the category of graded manifolds [3], [8], or one mimicks the usual definition of a manifold, having replaced Euclidean space with a suitable product of the odd and even parts of an exterior algebra as in the category of supermanifolds [6]. This paper establishes the equivalence of the category of supermanifolds with the category of graded manifolds.

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

  • [1] M. Batchelor, The structure of supermanifolds, Trans. Amer. Math. Soc. 253 (1979), 329-338. MR 536951 (80h:58002)
  • [2] F. Berezin, The method of second quantization, Academic Press, New York, 1966. MR 0208930 (34:8738)
  • [3] F. Berezin and G. Kac, Lie groups with commuting and anticommuting parameters, Math. USSR-Sb. 11 (1970), 311-326. MR 0265520 (42:429)
  • [4] F. Berezin and D. Leites, Supervarieties, Soviet Math. Dokl. 16 (1975), 1218-1222.
  • [5] L. Corwin, Y. Néeman and S. Sternberg, Graded Lie algebras in mathematics and physics (Bose-Fermi symmetry), Rev. Modern Phys. 47 (1975), 573-604. MR 0438925 (55:11828)
  • [6] B. de Witt, Differential supergeometry (in preparation).
  • [7] V. Kac, Lie superalgebras, Advances in Math. 26 (1977), 8-96. MR 0486011 (58:5803)
  • [8] B. Kostant, Graded manifolds, graded Lie theory and prequantization, Differential Geometric Methods in Mathematical Physics, Lecture Notes in Math., vol. 570, Springer-Verlag, Berlin and New York, 1977, pp. 177-306. MR 0580292 (58:28326)
  • [9] W. Rudin, Functional analysis, McGraw-Hill, New York, 1973. MR 0365062 (51:1315)
  • [10] M. Sweedler, Hopf algebras, Benjamin, New York, 1969. MR 0252485 (40:5705)
  • [11] F. Treves, Topological vector spaces, distributions and kernels, Academic Press, New York, 1967. MR 0225131 (37:726)

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Keywords: Supermanifolds, graded manifolds, graded Lie algebras, superspace, anticommuting coordinates
Article copyright: © Copyright 1980 American Mathematical Society

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