Notes for AKT-140224/0:27:53

Lie algebra of dimensions 1 and 2

1. one-dimensional Lie algebras are unique up to isomorphism. For if [math]\displaystyle{ \mathfrak{g} = \langle x \rangle }[/math] is a one dimensional Lie algebra, then since the bracket is antisymmetric, we have [math]\displaystyle{ [x, x] = 0 }[/math]. Thus the bracket is zero and [math]\displaystyle{ \mathfrak{g} }[/math] is unique up to isomorphism.

2. [math]\displaystyle{ \mathfrak{g} = \mathbb{F}^2 = \{ax + by \;|\; a, b \in \mathbb{F}\} }[/math]. [math]\displaystyle{ \mathfrak{g} }[/math] is a two-dimensional Lie algebra. There are only two of such up to isomorphism, that is, the one with the bracket equal to zero and the other with bracket [math]\displaystyle{ [x, y] = x }[/math].