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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, I


Authors: Y. Barnea and D. S. Passman
Journal: Trans. Amer. Math. Soc. 358 (2006), 1983-2010
MSC (2000): Primary 17B20, 17B70, 16W70
Published electronically: December 20, 2005
MathSciNet review: 2197439
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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
Email: passman@math.wisc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03986-3
PII: S 0002-9947(05)03986-3
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.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.