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Unformatted text preview: ence of a unit 1 is not assumed.
Deﬁnition 1.1.2. Let L be a vector space over a ﬁeld F . Then a bilinear operation
[··] : L × L → L sending (x, y ) to [xy ] is called a bracket if it satisﬁes the following two
[L1 ] [xx] = 0 for all x ∈ L.
[L2 ] (Jacobi identity ) [x[yz ]] + [y [zx]] + [z [xy ]] = 0 for all x, y, z ∈ L.
A vector space L with a bracket [··] is called a Lie algebra. This is an example of a
Let us analyze the two conditions. Condition [L1] implies:
[L1 ] [xy ] = −[yx] for all x, y ∈ L. (Bracket is skew commutative.)
Proof: [(x + y )(x + y )] = 0 = [xx] + [xy ] + [yx] + [yy ] = [xy ] + [yx].
Conversely, if the characteristic of the...
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- Fall '11