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Unformatted text preview: V → F is a bilinear form. Then the set of all
x ∈ gl(V ) so that
f (x(v ), w) + f (v, x(w)) = 0
for all v, w ∈ V is a Lie subalgebra of gl(V ) which we denote o(V, f )
Proof. It is clear that o(V, f ) is a vector subspace since the deﬁning equation is linear in
x. The following calculation shows that it is closed under Lie bracket.
f (xy (v ), w) + f (y (v ), x(w)) = 0
f (yx(v ), w) + f (x(v ), y (w)) = 0
f (v, xy (w)) + f (x(v ), y (w)) = 0
f (v, yx(w)) + f (y (v ), x(w)) = 0
If we take the alternating sum (+ - + -) of these equations we see that
f ([xy ](v ), w) + f (v, [xy ](w)) = 0
4 MATH 223A NOTES 2011 LIE ALGEBRAS Example 1.2.6. Particular examples of the above deﬁnition are as follows.
(1) Suppose that f is a nondegenerate symme...
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This document was uploaded on 03/09/2014 for the course MATH 223a at Brandeis.
- Fall '11