This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 crossproduct: [ 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....
View
Full
Document
This note was uploaded on 07/08/2011 for the course MAT 6932 taught by Professor Staff during the Summer '10 term at University of Florida.
 Summer '10
 Staff
 Algebra, Vector Space

Click to edit the document details