Editing Lie algebra (section)

=== Two dimensions ===
Some Lie algebras of low dimension are described here. See the [[classification of low-dimensional real Lie algebras]] for further examples.

* There is a unique nonabelian Lie algebra <math>\mathfrak{g}</math> of dimension 2 over any field ''F'', up to isomorphism.<ref>{{harvnb|Erdmann|Wildon|2006|loc=Theorem 3.1.}}</ref> Here <math>\mathfrak{g}</math> has a basis <math>X,Y</math> for which the bracket is given by <math> \left [X, Y\right ] = Y</math>. (This determines the Lie bracket completely, because the axioms imply that <math>[X,X]=0</math> and <math>[Y,Y]=0</math>.) Over the real numbers, <math>\mathfrak{g}</math> can be viewed as the Lie algebra of the Lie group <math>G=\mathrm{Aff}(1,\mathbb{R})</math> of [[Affine group|affine transformations]] of the real line, <math>x\mapsto ax+b</math>.

:The affine group ''G'' can be identified with the group of matrices
::<math> \left( \begin{array}{cc} a & b\\ 0 & 1 \end{array} \right) </math>
:under matrix multiplication, with <math>a,b \in \mathbb{R} </math>, <math>a \neq 0</math>. Its Lie algebra is the Lie subalgebra <math>\mathfrak{g}</math> of <math>\mathfrak{gl}(2,\mathbb{R})</math> consisting of all matrices
::<math> \left( \begin{array}{cc} c & d\\ 0 & 0 \end{array}\right). </math>
:In these terms, the basis above for <math>\mathfrak{g}</math> is given by the matrices
::<math> X= \left( \begin{array}{cc} 1 & 0\\ 0 & 0 \end{array}\right), \qquad Y= \left( \begin{array}{cc} 0 & 1\\ 0 & 0 \end{array}\right). </math>

:For any field <math>F</math>, the 1-dimensional subspace <math>F\cdot Y</math> is an ideal in the 2-dimensional Lie algebra <math>\mathfrak{g}</math>, by the formula <math>[X,Y]=Y\in F\cdot Y</math>. Both of the Lie algebras <math>F\cdot Y</math> and <math>\mathfrak{g}/(F\cdot Y)</math> are abelian (because 1-dimensional). In this sense, <math>\mathfrak{g}</math> can be broken into abelian "pieces", meaning that it is solvable (though not nilpotent), in the terminology below.