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lecture notes1 - MATH 223A NOTES 2011 LIE ALGEBRAS Contents...

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MATH 223A NOTES 2011 LIE ALGEBRAS Contents 1. Basic Concepts 2 1.1. Defnition 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. Defnition oF Lie algebra. Defnition 1.1.1. By an (nonassociative) algebra over a feld 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. Defnition 1.1.2. Let L be a vector space over a feld F . Then a bilinear operation [ ·· ]: L × L L sending ( x, y )to[ xy ] is called a bracket iF it satisfes the Following two conditions. [L1][ xx ] = 0 For all x L . [L2]( Jacobi identity )[ x [ ]] + [ 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 yy ]=[ xy ]. Conversely, iF the characteristic oF the feld 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 [ ]] = [[ xy ] z ] + [[ ] y ] . The term [[ ] 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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lecture notes1 - MATH 223A NOTES 2011 LIE ALGEBRAS Contents...

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