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12 examples example 121 the simplest example of a lie

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Unformatted text preview: orphism of Lie algebras is a linear map ϕ : L → L￿ so that ϕ([xy ]) = [ϕ(x)ϕ(y )] for all x, y ∈ L. 1.2. Examples. Example 1.2.1. The simplest example of a Lie algebra is given by letting [xy ] = 0 for all x, y ∈ L where L is any vector space over F . All conditions are clearly satisfied. A Lie algebra satisfying this condition (usually written as [LL] = 0) is called abelian. MATH 223A NOTES 2011 LIE ALGEBRAS 3 The word “abelian” comes from one standard interpretation of the bracket. Suppose that A is an associative algebra over F . Then the commutator [xy ] is defined by [xy ] = xy − yx. This is easily seen to be a bracket and is also called the Lie bracket of the associative algebra. Example 1.2.2. Suppose that V is any vect...
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This document was uploaded on 03/09/2014 for the course MATH 223a at Brandeis.

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