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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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Filtrations in semisimple Lie algebras, I
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by Y. Barnea and D. S. Passman PDF
Trans. Amer. Math. Soc. 358 (2006), 1983-2010 Request permission

Abstract:

In this paper, we study the maximal bounded $\mathbb {Z}$-filtrations of a complex semisimple Lie algebra $L$. Specifically, we show that if $L$ is simple of classical type $A_n$, $B_n$, $C_n$ or $D_n$, then these filtrations correspond uniquely to a precise set of linear functionals on its root space. We obtain partial, but not definitive, results in this direction for the remaining exceptional algebras. Maximal bounded filtrations were first introduced in the context of classifying the maximal graded subalgebras of affine Kac-Moody algebras, and the maximal graded subalgebras of loop toroidal Lie algebras. Indeed, our main results complete this classification in most cases. Finally, we briefly discuss the analogous question for bounded filtrations with respect to other Archimedean ordered groups.
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Additional Information
  • Y. Barnea
  • Affiliation: Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom
  • Email: y.barnea@rhul.ac.uk
  • D. S. Passman
  • Affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706
  • MR Author ID: 136635
  • Email: passman@math.wisc.edu
  • Received by editor(s): February 4, 2004
  • Published electronically: December 20, 2005
  • Additional Notes: The first author’s research was carried out while visiting the University of Wisconsin-Madison, Imperial College and the University of Kent. He thanks all three mathematics departments.
    The second author’s research was supported in part by NSA grant 144-LQ65.
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 358 (2006), 1983-2010
  • MSC (2000): Primary 17B20, 17B70, 16W70
  • DOI: https://doi.org/10.1090/S0002-9947-05-03986-3
  • MathSciNet review: 2197439