Very nilpotent basis and n-tuples in Borel subalgebras

Details Very nilpotent basis and n-tuples in Borel subalgebras Very nilpotent basis and n-tuples in Borel subalgebras

2010-10-05

arxiv.org/abs/1010.0821v2

Mathematics

Representation Theory

short note, 4 pages

Scientific paper

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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