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Very nilpotent basis and n-tuples in Borel subalgebras

Abstract : A (vector space) basis B of a Lie algebra is said to be very nilpotent if all the iterated brackets of elements of B are nilpotent. In this note, we prove a refinement of Engel's Theorem. We show that a Lie algebra has a very nilpotent basis if and only if it is a nilpotent Lie algebra. When g is a semisimple Lie algebra, this allows us to define an ideal of S((g^n)^*)^G whose associated algebraic set in g^n is the set of n-tuples lying in a same Borel subalgebra.
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Submitted on : Monday, January 16, 2012 - 3:13:01 PM
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Michael Bulois. Very nilpotent basis and n-tuples in Borel subalgebras. Comptes Rendus. Mathématique, Académie des sciences (Paris), 2011, 349 (3-4), pp.149-152. ⟨10.1016/j.crma.2010.12.005⟩. ⟨hal-00660382⟩



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