# 21a - Lecture 21 Nilpotent Lie Algebras Definition 1...

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Unformatted text preview: Lecture 21 Nilpotent Lie Algebras Definition 1. The (lower) central series of g is ( g n : n ≥ 0) defined by g = g g n +1 = [ g , g n ] Definition 2. Say g is nilpotent iff g n = 0 for some n Example 1. • Abelian ⇒ nilpotent ( n = 1) • Nilpotent ⇒ Solvable [ g ( n ) ⊂ g n by induction] • Solvable notdblarrowright Nilpotent [ 2 d nonabelian: basis x, y with bracket [ x, y ] = x g 1 = g (1) = [ g , fg ] = F x g 2 = F x In fact g n = F x negationslash = 0 ∀ n ≥ 1 ] Theorem 1. A Lie algebra is not nilpotent iff it contains x, y with [ x, y ] = x . Proof. . ( ⇐ ) g n ⊃ F x ∀ n ≥ 1 ( ⇒ ) By Engel (See below), some z ∈ g is not ad-nilpotent. Thus the decreasing sequence ad z ( g ) ⊇ ad 2 z ( g ) ⊇ . . . 1 stabilizes at, say ad n z ( g ) = ad n +1 z negationslash = 0. Now ad z : ad n z ( g ) → ad n z ( g ) is a linear isomorphism. Chose nonzero eigenvalue λ with eigenvector x . Then ad z ( x ) = λx . Finally Let y = − 1 λ z Example 2. The 3 d Heisenberg algebra (basis...
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21a - Lecture 21 Nilpotent Lie Algebras Definition 1...

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