Group.

Group is defined by its structure constants $f^{\alpha}_{\beta \gamma}$ which are anti-symmetric $( f^{\alpha}_{\beta \gamma} = -f^{\alpha}_{\gamma \beta})$ and obey the Jacoby identity:

$$f^{\delta}_{\alpha \epsilon}f^{\epsilon}_{\beta \gamma} + f^{... ...f^{\delta}_{\beta \epsilon}f^{\epsilon}_{\gamma \alpha} = 0\;.$$

Group generators are Hermitian matrices $\hat\tau_{\alpha}$ which satisfy the commutation relations:

$$\hat\tau_{\alpha}\hat\tau_{\beta} - \hat\tau_{\beta}\hat\tau_{\alpha}= i\,f^{\gamma}_{\alpha \beta} \hat\tau_{\gamma}\;.$$

In particular the generators in the adjoint representation are

$$(\hat\tau_{\alpha})^i_j = -i\,f^i_{j\alpha}\;.$$

Group transformation may be represented with the help of group generators as

$$\hat g(w)=exp(i\hat\tau_{\alpha}\omega^{\alpha})\;.$$

We assume that the Killing metric is orthonormal:

$$-\, \frac{1}{2} f^{\epsilon}_{\gamma \alpha} f^{\gamma}_{\epsilon \beta} = \delta_{\alpha \beta}\;.$$

This metric allows one to raise and lower the group indices. In the case of orthonormal Killing metric the structure constants are fully antisymmetric under interchange of any pair of indices.