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Journal of Lie Theory 20 (2010), No. 1, 049--063
Copyright Heldermann Verlag 2010

On the Index of the Quotient of a Borel Subalgebra by an ad-Nilpotent Ideal

Céline Righi
Dip. di Matematica, Istituto G. Castelnuovo, Università di Roma "La Sapienza", Piazzale Aldo Moro 5, 00185 Rome, Italy

Rupert W. T. Yu
Dép. de Mathématiques, Université de Poitiers, Téléport 2 - BP 30179, Blvd Marie et Pierre Curie, 86962 Futuroscope Chasseneuil, France

We give upper bounds for the index of the quotient of a Borel subalgebra of a simple Lie algebra or its nilpotent radical by an ad-nilpotent ideal. For the nilpotent radical quotient, our bound is a generalization of the formula for the index given by Panov in the type A case. In general, this bound is not exact. Using results of Panov ["On the index of certain nilpotent Lie algebras", J. of Math. Sci. 161 (2009) 122--129], we show that the upper bound for the Borel quotient is exact in the type A case, and we conjecture that it is exact in general.

Keywords: Index, Borel subalgebras, ad-nilpotent ideals.

MSC: 17B08, 17B20, 17B22

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