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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Filtrations in semisimple Lie algebras, II


Authors: Y. Barnea and D. S. Passman
Journal: Trans. Amer. Math. Soc. 360 (2008), 801-817
MSC (2000): Primary 17B20, 17B70, 16W70
Published electronically: September 18, 2007
MathSciNet review: 2346472
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Abstract: 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: y.barnea@rhul.ac.uk

D. S. Passman
Affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706
Email: passman@math.wisc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04224-9
PII: S 0002-9947(07)04224-9
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.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.