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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, II
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by Y. Barnea and D. S. Passman PDF
Trans. Amer. Math. Soc. 360 (2008), 801-817 Request permission


In this paper, we continue our study of the maximal bounded $\mathbb {Z}$-filtrations of a complex semisimple Lie algebra $L$. Specifically, we discuss the functionals which give rise to such filtrations, and we show that they are related to certain semisimple subalgebras of $L$ of full rank. In this way, we determine the “order” of these functionals and count them without the aid of computer computations. The main results here involve the Lie algebras of type $E_6$, $E_7$ and $E_8$, since we already know a good deal about the functionals for the remaining types. Nevertheless, we reinterpret our previous results into the new context considered here. Finally, we describe the associated graded Lie algebras of all of the maximal filtrations obtained in this manner.
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Additional Information
  • Y. Barnea
  • Affiliation: Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom
  • Email:
  • D. S. Passman
  • Affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706
  • MR Author ID: 136635
  • Email:
  • Received by editor(s): August 10, 2004
  • Received by editor(s) in revised form: October 19, 2005
  • Published electronically: September 18, 2007
  • Additional Notes: The first author’s research was carried out while visiting the departments of Mathematics at Imperial College and at the University of Kent. He thanks both departments.
    The second author’s research was supported in part by NSA grant 144-LQ65.
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 801-817
  • MSC (2000): Primary 17B20, 17B70, 16W70
  • DOI:
  • MathSciNet review: 2346472