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01 - Lecture 01 Denition 1 A Lie algebra is a vector space...

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Lecture 01 Definition 1. A Lie algebra is a vector space g equipped with a bilinear product [ , ] : g × g g that is alternating: w g [ w, w ] = 0 and satisfies the Jacobi identity: if x, y, z g then [ x, [ y, z ]] + [ y, [ z, x ]] + [ z, [ x, y ]] = 0 . Here, the underlying scalar field may be arbitrary, but we shall usually take it to be the real numbers R or the complex numbers C . Notice that the alternating property implies the property x, y g [ x, y ] + [ y, x ] = 0 and that the two properties are equivalent in characteristic other than two. Example 1. g = R 3 with bracket defined by cross-product: [ a, b ] := a × b. Only the Jacobi identity is less than obvious: for this, use the familiar ‘vector triple product identity’ c × ( a × b ) = ( b · c ) a - ( c · a ) b. Example 2. Let A be any algebra with associative product (indicated simply by juxtaposition). The ‘commutator bracket’ defined by X, Y A [ X, Y ] = XY - Y X converts A into a Lie algebra. Again, only the Jacobi identity calls for any work in verification; this should be carried out just once as an easy exercise.
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