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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 .

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

Abstract:

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.
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Additional Information
  • © Copyright 1979 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 253 (1979), 329-338
  • MSC: Primary 58A05; Secondary 83E99
  • DOI: https://doi.org/10.1090/S0002-9947-1979-0536951-0
  • MathSciNet review: 536951