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MATH 223A NOTES 2011 LIE ALGEBRAS Contents 1. Basic Concepts 2 1.1. Definition of Lie algebra 2 1.2. Examples 2 1.3. Derivations 4 1.4. Abstract Lie algebras 4 Date : September 6, 2011. 1
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2 MATH 223A NOTES 2011 LIE ALGEBRAS 1. Basic Concepts 1.1. Definition of Lie algebra. Definition 1.1.1. By an (nonassociative) algebra over a field F we mean a vector space A together with an F -bilinear operation A × A A which is usually written ( x, y ) xy . The adjective “nonassociative” means “not necessarily associative”. An associative algebra is an algebra A whose multiplication rule is associative: x ( yz ) = ( xy ) z for all x, y, z A . The existence 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 conditions. [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 nonassociative algebra. 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 field F is not equal to 2 then [L1 ] implies that 2[ xx ] = 0 implies [L1]. So, [L1] and [L1 ] are equivalent when char F = 2. The second condition [L2] can be rewritten as follows: [ x [ yz ]] = [[ xy ] z ] + [[ zx ] y ] . The term [[ zx ] y ] prevents L from being associative. Since z, x, y are arbitrary we obtain: Proposition 1.1.3. A Lie algebra is associative if and only if [[ LL ] L ] = 0 .
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