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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 welldefined. 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 φ : g → h that preserves brackets: x,y ∈ g ⇒ φ ([ x,y ]) = [ φ ( x ) ,φ ( y )] . An isomorphism from g to h is a homomorphism φ : g → h for which there is a homomor phism ψ : h → g such that both ψ ◦ φ = id g and φ ◦ ψ = id h (alternatively, φ is a bijective homomorphism).is a bijective homomorphism)....
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 Summer '10
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 Algebra, Vector Space

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