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Lie Groups

  1. Feb 23, 2010 #1
    Consider R^4 with nondegenerate inner product [itex] \left( x,y \right)= x^T \eta y[/itex] where [itex]\eta=\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right)[/itex]
    Let [itex]e_1,e_2,e_3,e_4[/itex] be the basis of R^4 with [itex](e_i,e_i)=\begin{cases} 1 \quad i=1,2,3 \\ -1 \quad i=4 \end{cases}[/itex]
    Let [itex]P \subset Aff(\mathbb{R}^4)[/itex] be the subset of the affine group that preserves [itex]\delta(x,y)=(x-y)^T \eta (x-y)[/itex]

    we have to show that P is a linear group and determine its lie algebra [itex]\mathfrak{p}[/itex] as a Lie subalgebra of [itex]gl(5,\mathbb{R})[/itex] and then find the dimension of [itex]\mathfrak{p}[/itex].

    to show its a linear group:

    i guess i need to establish that its a subset of invertible matrices (these will be 2x2 as then it would be isomorphic to R^4 i think) but i dont really know how to show this explicitly
    then given A in P
    [itex]\delta(Ax,Ay)=(Ax-Ay)^T \eta (Ax-Ay) = (x-y)^T A^T \eta A (x-y) = delta(x,y)[/itex] iff [itex]A^T \eta A = \eta[/itex]. (this is what elements of P must satisfy)

    then given a particular A in P, consider A^{-1}

    [itex]\delta(A^{-1}x-A^{-1}y)^T \eta (A^{-1}x-A^{-1}y)=(x-y)^T(A^{-1})^T \eta A^{-1} (x-y)[/itex]
    but [itex](A^{-1})^T \eta A^{-1}=(A^T)^{-1} \eta A^{-1} = A^{T}^{-1} \eta^{-1} A^{-1}=A^T \eta A = \eta [/itex]where we used the property [itex]\eta^{-1}=\eta[/itex]
    this means that [itex]A^{-1} \in P[/itex]
    similarly if A,B are in P

    [itex] \delta( ABx,ABy ) = (x-y)^T B^T A^T \eta AB (x-y) = (x-y)^T B^T \eta B (x-y) = (x-y)^T \eta (x-y) \Rightarrow AB \in P[/itex]
    therefore P is a linear group. is this correct?

    then for the next bit i said let [itex]A= exp ( \tau X) \in P[/itex]
    we know [itex]A^T \eta A = \eta[/itex]
    therefore [itex] exp ( \tau X^T ) \eta exp( \tau X ) = \eta[/itex]

    differentiating wrt tau and evaluating at tau=0 we get
    [itex]X=- \eta^{-1} X^T \eta[/itex]
    which implies
    [itex]\mathfrak{p} := \{ X \in Mat ( 2 , \mathbb{R} ) | X=-\eta^{-1} X^T \eta \}[/itex]
    is this right? i guess this is determining p but how do i determine it as a lie subalgebra of gl(5,R) - i dont even know what that means tbh!
    and what about the dimension?

  2. jcsd
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