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Licensed Unlicensed Requires Authentication Published by De Gruyter September 10, 2018

The even, the odd, the superalgebras and their derivations

  • Claude Roger EMAIL logo

Abstract

We give an introduction to superalgebra, founded on the difference between even (commuting) and odd (anti-commuting) variables. We give a sketch of Graßmann’s work, and show how derivations of those structures induce various superalgebra structures, Lie superalgebras of Cartan type being obtained with even derivations, while odd derivations induce Jordan-type superalgebras.

MSC 2010: 17B60; 17B65; 17B68; 81Q60

References

[1] R. Beekes, Comparative Indo-European Linguistics, John Benjamins, Amsterdam, 2011. 10.1075/z.172Search in Google Scholar

[2] P. Deligne and J. W. Morgan, Notes on supersymmetry (following Joseph Bernstein), Quantum Fields and Strings: A Course for Mathematicians. Vol. 1, 2, American Mathematical Society, Providence (1999), 41–97. Search in Google Scholar

[3] J. Dieudonné, Abrégé d’histoire des mathématiques 1700–1900, Hermann, Paris, 1978. Search in Google Scholar

[4] L. Guieu and C. Roger, The Virasoro Algebra and Group. Geometric and Algebraic Aspects, Generalizations (in French), Les Publications CRM, Montreal, 2007. Search in Google Scholar

[5] D. A. Leĭtes, Introduction to the theory of supermanifolds, Uspekhi Mat. Nauk 35 (1980), no. 1(211), 3–57. 10.1070/RM1980v035n01ABEH001545Search in Google Scholar

[6] A. Martinet, Des steppes aux océans, L’indo-européen et les “Indo-Européens”, Bibliothèque Scientifique Payot, Payot, 1994. Search in Google Scholar

[7] K. McCrimmon, A Taste of Jordan Algebras, Universitext, Springer, New York, 2004. Search in Google Scholar

[8] A. Meillet, Aperçu d’une histoire de la langue grecque, J. Savants 11 (1913), no. 6, 275–280. 10.1017/CBO9780511706219Search in Google Scholar

[9] V. Ovsienko, Lie antialgebras: Prémices, J. Algebra 325 (2011), 216–247. 10.1016/j.jalgebra.2010.10.003Search in Google Scholar

[10] V. Ovsienko and C. Roger, Looped cotangent Virasoro algebra and non-linear integrable systems in dimension 2+1, Comm. Math. Phys. 273 (2007), no. 2, 357–378. 10.1007/s00220-007-0237-zSearch in Google Scholar

[11] H.-J. Petsche, Graßmann, Vita Math. 13, Birkhäuser, Basel, 2006. Search in Google Scholar

[12] H.-J. Petsche, L. Kannenberg, G. Keßler and J. Liskowacka, Hermann Graßmann—Roots and Traces, Birkhäuser, Basel, 2009. 10.1007/978-3-0346-0155-9Search in Google Scholar

Received: 2018-05-14
Accepted: 2018-07-27
Published Online: 2018-09-10
Published in Print: 2018-10-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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