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Structure and Representation Theory of Kac–Moody Superalgebras

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Highlights in Lie Algebraic Methods

Part of the book series: Progress in Mathematics ((PM,volume 295))

Abstract

The aim of these lecture notes is to give an introduction to the structure and representation theory of Lie superalgebras.

We start by reviewing some basic facts about finite-dimensional and Kac–Moody Lie superalgebras. Then we review the classification of Kac–Moody superalgebras of finite growth and give a survey of results about highest weight integrable representations of Kac–Moody Lie superalgebras.

In the last section we discuss the representation theory of finite-dimensional superalgebras. We formulate an analogue of a theorem of Harish-Chandra and review some geometric methods: associated variety and the Borel–Weil–Bott theorem. We omit all long and technical proofs referring to the original papers but try to explain the main ideas.

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Notes

  1. 1.

    Here we list only exceptional Lie superalgebras \(\mathfrak{g}=\mathfrak{g}_{0}+\mathfrak{g}_{1}\) with \(\mathfrak{g}_{1}\ne 0\).

  2. 2.

    We modify Kac’s original notation in order to avoid confusion with the usual Lie algebra F 4.

  3. 3.

    W(0|n) is simple if n≥2.

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Acknowledgements

I thank Sergey Shashkov and Alexander Shapiro for preparing a preliminary version of the manuscript based on the notes they took during the lectures.

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Authors and Affiliations

  1. Department of Mathematics, UC Berkeley, Berkeley, CA, 94720, USA

    Vera Serganova

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  1. Vera Serganova
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Correspondence to Vera Serganova .

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Editors and Affiliations

  1. Department of Mathematics, Weizmann Institute, Rehovot, 76100, Israel

    Anthony Joseph

  2. Department of Mathematics, Univeristy of Haifa, Haifa, 31905, Israel

    Anna Melnikov

  3. School of Engineering and Science, Jacobs University, Bremen, 28759, Germany

    Ivan Penkov

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Serganova, V. (2012). Structure and Representation Theory of Kac–Moody Superalgebras. In: Joseph, A., Melnikov, A., Penkov, I. (eds) Highlights in Lie Algebraic Methods. Progress in Mathematics, vol 295. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8274-3_3

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