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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.
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
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.
- Fall '11