02 - Lecture 02 Suppose k is an ideal of g . The...

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Unformatted text preview: Lecture 02 Suppose k is an ideal of g . The corresponding quotient algebra is constructed as follows: Let g / k be the quotient vector space. Define a bracket [ , ] on g / k by [ x + k ,y + k ] := [ x,y ] + k . This bracket is well-defined. Indeed, suppose x + k = x + k and y + k = y + k . Then [ x ,y ]- [ x,y ] = [ x- x,y ] + [ x,y- y ] . Since k is an ideal, each of [ x- x,y ] and [ x,y- y ] is in k . Since k is a subspace, [ x- x,y ] + [ x,y- y ] k . It is an easy exercise to verify that [ , ] defined as above on g / k is bilinear, alternating and satisfies the Jacobi identity. Example 1. The quotient of g by its derived algebra [ g , g ] is abelian (this is obvious from the definition of derived algebra). In fact, [ g , g ] is the smallest ideal k for which g / k is abelian: If g / k is abelian, then for each x,y g , [ x,y ] + k = [ x + k ,y + k ] = 0 so that [ x,y ] k . By considering the linear span of all x,y g , we see that [ g , g ] k . Example 2. Z ( g /Z ( g )) need not be zero. Example? See the Heisenberg algebra (next lecture). Definition 1. A homomorphism from g to h is a linear map : g h that preserves brackets: x,y g ([ x,y ]) = [ ( x ) , ( y )] . An isomorphism from g to h is a homomorphism : g h for which there is a homomor- phism : h g such that both = id g and = id h (alternatively, is a bijective homomorphism).is a bijective homomorphism)....
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02 - Lecture 02 Suppose k is an ideal of g . The...

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