Transactions of the American Mathematical Society

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



Left-symmetric algebras for ${\mathfrak{gl}}(n)$

Author: Oliver Baues
Journal: Trans. Amer. Math. Soc. 351 (1999), 2979-2996
MSC (1991): Primary 55N35, 55Q70, 55S20
Published electronically: March 8, 1999
MathSciNet review: 1608273
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Abstract: We study the classification problem for left-symmetric algebras with commutation Lie algebra ${\mathfrak{gl}}(n)$ in characteristic $0$. The problem is equivalent to the classification of étale affine representations of ${\mathfrak{gl}}(n)$. Algebraic invariant theory is used to characterize those modules for the algebraic group $\operatorname{SL}(n)$ which belong to affine étale representations of ${\mathfrak{gl}}(n)$. From the classification of these modules we obtain the solution of the classification problem for ${\mathfrak{gl}}(n)$. As another application of our approach, we exhibit left-symmetric algebra structures on certain reductive Lie algebras with a one-dimensional center and a non-simple semisimple ideal.

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Additional Information

Oliver Baues
Affiliation: Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf, Universitäts- strasse 1, D-40225 Düsseldorf, Germany
Address at time of publication: Department of Mathematics, ETH-Zentrum, CH-8092 Zürich, Switzerland

Received by editor(s): February 10, 1997
Published electronically: March 8, 1999
Article copyright: © Copyright 1999 American Mathematical Society