10a - Lecture 10 We show that there are infinitely many...

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
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lecture 10 We show that there are infinitely many non-isomorphic lie algebras g such that Dim g = 3 and Dim[ g , g ] = 2. We start by fixing an element z ∈ g- [ g , g ], and examining the map ad z : [ g , g ] → [ g , g ]. We have already seen that [ g , g ] is abelian, and that ad z is a bijection from [ g , g ] to itself. We split into two cases. Case 1 ad z is not diagonalizable, in which case we find that g is unique up to isomorphism. Case 2 ad z is diagonalizable. In this case choose a basis B = { x,y } for [ g , g ] such that ad z ( x ) = λx and ad z ( y ) = μy for some λ,μ ∈ F . Claim 1. Let g and g be as in Case 2, with corresponding eigenvalues λ,μ and λ ,μ respectively. Then g ≡ g if and only if { λ μ , μ λ } = { λ μ , μ λ } . Proof. First, we show that for a lie algebra g , the set { λ μ , μ λ } is uniquely deter- mined by g . Let w ∈ g- [ g , g ], and consider ad x . We see that w = ax + by + cz , with c 6 = 0. Therefore, since [ g , g ] is abelian, we have [ ax + by + cz,x ] = [ cz,x ] = cλx and [ ax + by + cz,y ] = [ cz,y ] = cμy. Therefore, x and y are eigenvalues with corresponding eigenvalues cλ and cμ . This shows that the set of ratios of eigenvalues { cλ cμ , cμ cλ } = { λ μ , μ λ } is uniquely determined by g . Now, suppose that φ : g → g is a lie algebra isomorphism. Then since φ (ad z ( v )) = ad φ ( z ) we see that φ ◦ ad z = ad φ ( z ) ◦ φ , which implies that ad φ ( z ) = φ ◦ ad z ◦ φ- 1 . Therefore, since ad z and ad φ ( z ) are conjugate, they have the same eigenvalues and hence { λ μ , μ λ } = { λ μ , μ λ } ....
View Full Document

{[ snackBarMessage ]}

Page1 / 5

10a - Lecture 10 We show that there are infinitely many...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online