Lie subalgebra

Generators and dimension

The dimension of a Lie algebra over a field means its dimension as a vector space. In physics, a vector space basis of the Lie algebra of a Lie group ${\displaystyle G}$ may be called a set of generators for ${\displaystyle G}$. (They are "infinitesimal generators" for ${\displaystyle G}$, so to speak.) In mathematics, a set ${\displaystyle S}$ of generators for a Lie algebra ${\displaystyle {\mathfrak {g}}}$ means a subset of ${\displaystyle {\mathfrak {g}}}$ such that any Lie subalgebra (as defined below) that contains ${\displaystyle S}$ must be all of ${\displaystyle {\mathfrak {g}}}$. Equivalently, ${\displaystyle {\mathfrak {g}}}$ is spanned (as a vector space) by all iterated brackets of elements of ${\displaystyle S}$.