One interesting example: For the one-dimensional quantum simple harmonic oscillator, the operators $1$, $\hat{x}$ and $\hat{p}$ generate a real Lie algebra (the Heisenberg algebra). However, it is often useful to work instead in terms of the raising and lowering operators, so that our generators are $1, \hat{a},\hat{a}^\dagger$. Since $\hat{a} = \hat{x} + i\hat{p}$, the algebra they generate is the complexification of the Heisenberg algebra.

There is a useful sense in which the Lie algebra of any Lie group is "naturally" real, since it's a tangent space of a manifold. For example $GL_n(\mathbb{C})$, considered as an abstract Lie group, is a $2n^2$-dimensional manifold, so the Lie algebra is "really" a real vector space with $2n^2$ basis vectors. In physics we usually think of it as a complex vector space with $n^2$ basis vectors, but we don't have to. Every Lie algebra I can think of in physics comes from some Lie group, so complex Lie algebras come only from either complexification (as in the raising/lowering operator case) or from treating a $2n$-dimensional real Lie algebra as an $n$-dimensional complex Lie algebra. (I don't have a name for the latter process, or much understanding of it - suggestions welcome).

This post imported from StackExchange Physics at 2020-12-04 11:35 (UTC), posted by SE-user Daniel