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22 suppose that x is an element of a lie algebra l so

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Unformatted text preview: (x, y ) where µ : A × A → A is multiplication. Then δ ◦ µ = µ ◦ (δ1 + δ2 ) where δ1 (x, y ) = (δ x, y ), δ2 (x, y ) = (x, δ y ) Since δ1 , δ2 commute, we can use the lemma to get: exp δ (xy ) = exp δ ◦ µ(x, y ) = µ ◦ exp(δ1 + δ2 )(x, y ) = µ ◦ exp(δ1 )exp(δ2 )(x, y ) = exp δ (x) exp δ (y ) Definition 2.2.2. Suppose that x is an element of a Lie algebra L so that adx is nilpotent. Then exp adx is called an inner automorphism of L. Proposition 2.2.3. Suppose that L is the Lie algebra of an associative algebra A with unity 1. Let x ∈ A be a nilpotent element. Then: ￿ (1) exp x = xk /i! is a unit in A. (2) adx is a nilpotent endomorphism of L. (3) exp adx is conjugation by exp x. Proof. By the Lemma, exp(−x) is the inverse of exp x. The other statement follow from the following trick: Write adx = λx + ρ−x where λx is “left multiplication by x” and ρ−x is “right multiplication by −x”: λx (y ) = xy and ρ−x (y ) = −yx. Since left and right multiplication are commuting oper...
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This document was uploaded on 03/09/2014 for the course MATH 223a at Brandeis.

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