Structure constants in Lie Algebra

Question

As a nit-picking question, I wanted to clarify a point of confusion. This arises from definitions found in a plethora of books, lectures notes and even the Wikipage on structure constants and Lie algebras. See this link to the Wiki page.

A generic Lie algebra that defines a group is written in terms of the group generators $X^a$. The Lie algebra is

$$[X^a,X^b] = i f_{abc}X^c.$$

I am happy that the structure constant $f_{abc}$ is a real number, and that $f_{abc} = - f_{bac} $ by commutator properties.

My issue is this; in the $SO(3)$ rotations, the Lie algebra is in terms of AM operators $J_i$. Hence, the Lie algebra is now

$$[J_i,J_j] = i \varepsilon_{ijk}J_k.$$ However, I am a bit uncomfortable with why all the indices are down. My only explanation for this is that since the structure constant $f_{ijk} = \varepsilon_{ijk}$ in this case is a real number, one can freely write the indices as we wish. So my original statement for the general Lie algebra could be equally written as

$$[X^a,X^b] = i f^{abc}X^c$$

or in particular

$$[X_a,X_b] = i f_{ab}^{\,\,\,\, c}X_c = i f_{abc}X_c.$$

Is this the right way to think about this?

Answer 1

1. Given a basis $(t_1,\ldots t_n)$ for a Lie algebra $L$ the structure constants $f_{ab}{}^{c}$ satisfy $$ [t_a,t_b]~=~\sum_{c=1}^nf_{ab}{}^{c} t_c.\tag{1} $$

2. Given a non-degenerate symmetric bilinear form $\kappa: L\times L \to \mathbb{F}$, we can define a metric $\kappa_{ab}=\kappa(t_a,t_b)$ to raise and lower indices.

3. If the Lie algebra is semisimple, we can use the Killing form as $\kappa$. See also this related Phys.SE post.