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can some one give me an example of two-dimensional non-abelian Lie algebra?

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2 Answers 2

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Let $X=\begin{bmatrix} a&b\\0&0 \end{bmatrix} $ and $Y=\begin{bmatrix} x&y\\0&0 \end{bmatrix} $. Furthermore, let the Lie bracket be the matrix commutator: $[X,Y]:=XY-YX$. We get $XY= \begin{bmatrix} ax&ay\\0&0 \end{bmatrix}$, but $YX= \begin{bmatrix} ax&bx\\0&0 \end{bmatrix}$.

(That is the lie algebra of the affine group mentioned by Eric).

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thanks for your answer sir –  user46309 Nov 2 '12 at 2:52
    
i want ask a little question, how to define the bracket in Lie algebra of the two dimensional affine group in above so we can say it's defines a lie algebra, –  user46309 Nov 2 '12 at 8:29
    
As Eric said, the 1D (!) affine group consists of transformation of the form $x\rightarrow ax+b$. Written in homogeneous matrix form, these transformation looks like: $\begin{bmatrix} a & b\\0&1\end{bmatrix}$. –  Hauke Strasdat Nov 10 '12 at 3:37
    
In other words, the Lie group Aff(1) consists of matrices of the form $\begin{bmatrix} a & b\\0&1\end{bmatrix}$. The Lie Algebra aff(1) is the tangent space of Aff(1). It is spanned by the derivatives of Aff(1). The partial derivative with respect to $a$ is: $\begin{bmatrix} 1 & 0\\0&0\end{bmatrix}$, the one with respect to $b$ is $\begin{bmatrix} 0 & 1\\0&0\end{bmatrix}$. –  Hauke Strasdat Nov 10 '12 at 3:42
    
Thus the Lie algebra aff(1) is spanned by the basis vectors $\begin{bmatrix} 1 & 0\\0&0\end{bmatrix}$ and $\begin{bmatrix} 0 & 1\\0&0\end{bmatrix}$. Therefore, its element are of the general form $\begin{bmatrix} x & y\\0&0\end{bmatrix}$. Since Aff(1) is a matrix Lie group, the Lie bracket of aff(1) is the common matrix commutator as defined in the answer above. Does this answer your question? –  Hauke Strasdat Nov 10 '12 at 3:45

This is something easy to come up with: take a basis $\{X,Y\}$ of your space. Then to be non-abelian $[X,Y]$ has to be non-zero. So try $[X,Y] = X$. It's straightforward to verify this satisfies the axioms of a Lie algebra. With a little more work you can show this is the unique (up to isomorphism) two dimensional non-abelian Lie algebra.

This Lie algebra has a geometric interpretation as the Lie algebra of affine transformations of the real line, i.e. all maps $\mathbb R \to \mathbb R$ of the form $x \mapsto ax + b$, $a,b \in \mathbb R$.

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yeah i know this is the unique (up to isomorphism) two dimensional non-abelian Lie algebra. but can u show me example of two dimensional lie algebra(from what vector space to what vector space and lie bracket define in there). like if we talk about three dimensional lie algebra , then sl(2,C) is three dimensional algebra, like that btw thanks for your answer –  user46309 Oct 29 '12 at 15:59
    
It's the Lie algebra of the two-dimensional affine group, which is the semi-direct product of $GL(1,\mathbb R) \simeq \mathbb R^*$ with $(\mathbb R, +)$. –  Eric O. Korman Oct 29 '12 at 16:18
    
ok thanks for your answer –  user46309 Nov 2 '12 at 2:51
    
i want ask a little question, how to define the bracket in Lie algebra of the two dimensional affine group in above so we can say it's defines a lie algebra, –  user46309 Nov 2 '12 at 8:28

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