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Unformatted text preview: 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 anyinto a Lie algebra....
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