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Unformatted text preview: hat L is a subalgebra of n(n, F ) up to
isomorphism. (Assume dim V = n is ﬁnite.)
(3) Show that any nilpotent subalgebra of gl(V ) is isomorphic to a subalgebra of
n(n, F ) where n = dim V .
3.2. Lie’s theorem. When we go to solvable Lie algebras we need the ground ﬁeld F to
be algebraically closed of characteristic 0. So, we might as well assume that F = C.
Then we have the following theorem whose statement and proof are similar to the
statement and proof of the key lemma for Engel’s Theorem.
Theorem 3.2.1 (Lie). Suppose that L ⊆ gl(V ) is a solvable linear Lie algebra over C.
Then there exists a nonzero vector v ∈ V which is a simultaneous eigenvector of every
element of L, i.e., x(v ) = λ(x)v for every x ∈ L where λ(x) ∈ C.
Remark 3.2.2. Before proving this we note that, the function λ : L → C is a linear map. (1) ax(v ) = aλ(x)v . So, λ(ax) = aλ(x).
(2) (x + y )(v ) = x(v ) + y (v ) = λ(x)v + λ(y )v = (λ(x) + λ(y ))(v ). So, λ(x + y ) =
λ(x) + λ(y ). MATH 223A NOTES 2011 LIE ALGEBRAS 11 Furthermore, note that if λ : L → C is a linear map then the equation x(v ) = λ(x)(v ) is
a linear equation in x. Therefore, if this equation holds for all x in a spanning set for L
then it holds for all x in L.
Proof. The proof is by induction on the dimension of L....
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This document was uploaded on 03/09/2014 for the course MATH 223a at Brandeis.
- Fall '11