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 ${\mathfrak {g}}=\langle x\rangle$ is a one dimensional Lie algebra, then since the bracket is antisymmetric, we have $[x,x]=0$. Thus the bracket is zero and ${\mathfrak {g}}$ is unique up to isomorphism.

2. ${\mathfrak {g}}=\mathbb {F} ^{2}=\{ax+by\;|\;a,b\in \mathbb {F} \}$. ${\mathfrak {g}}$ 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 $[x,y]=x$.