From Lie algebras to polar spaces

Jos in 't panhuis

We start with a simple Lie algebra g generated by a set of extremal elements E over a field k of characteristic not 2. Here, a nonzero element x of g is called extremal if [x, [x, g]] = k x.

If x, y are distinct extremal elements of g, then x and y generate a Lie subalgebra isomorphic to either the 2-dimensional abelian Lie algebra, the 3-dimensional Heisenberg algebra h, or the 3-dimensional Lie algebra sl2. We assume that the second case does not occur in g. Moreover, we write k x k y to denote that two extremal elements x and y generate the 2-dimensional abelian Lie algebra.

We write P(E) = { k x | x E } and L(E)={ ({X, Y}) | X, Y P(E) X Y } and prove that (P(E) , L(E)) is a polar space.

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