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Bulletin of the London Mathematical Society Advance Access originally published online on November 26, 2008
Bulletin of the London Mathematical Society 2009 41(1):1-15; doi:10.1112/blms/bdn087
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© 2008 London Mathematical Society

Rigid representations of a double quiver of type A, and Richardson elements in seaweed Lie algebras

Bernt Tore Jensen

Department of Mathematical Sciences
Norwegian University of Science and Technology
N-7034 Trondheim
Norway
berntj@math.ntnu.no

Xiuping Su

Mathematisches Insitut
Universität zu Köln
Weyertal 86-90
50931 Köln
Germany
xsu@math.uni-koeln.de

Rupert Wei Tze Yu

UMR 6086 CNRS
Département de Mathématiques
Université de Poitiers
Téléport 2 – BP 30179
Boulevard Marie et Pierre Curie
86962 Futuroscope Chasseneuil cedex
France

Received 24 July 2007. Revision received 24 June 2008.

In this paper, we show that there is always an open adjoint orbit in the nilpotent radical of a seaweed Lie algebra in gln(k), thus answering positively in this gln(k) case to a question raised independently by Michel Duflo and Dmitri Panyushev. The proof gives an explicit construction, using {Delta}-filtered modules of quasi-hereditary algebras arising from quotients of the double of quivers of type A. An example of a seaweed Lie algebra in a simple Lie algebra of type E8 not admitting an open orbit in its nilpotent radical is given.

2000 Mathematics Subject Classification 16G20, 17B20.

The first author is supported by the NFR project HoGeMetAlg, and the second author is supported by the Marie-Curie Fellowship IIF.


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