Invariant tensors for simple groups

1997/06/03 by J. A. de Azcarraga, A. J. Macfarlane, A. J. Mountain +1 · 2 voices · 1 citation
#math-ph #hep-th #math.QA

paper · pdf · doi:10.1016/s0550-3213(97)00609-3

Abstract

The forms of the invariant primitive tensors for the simple Lie algebras A_l, B_l, C_l and D_l are investigated. A new family of symmetric invariant tensors is introduced using the non-trivial cocycles for the Lie algebra cohomology. For the A_l algebra it is explicitly shown that the generic forms of these tensors become zero except for the l primitive ones and that they give rise to the l primitive Casimir operators. Some recurrence and duality relations are given for the Lie algebra cocycles. Tables for the 3- and 5-cocycles for su(3) and su(4) are also provided. Finally, new relations involving the d and f su(n) tensors are given.

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Discussions

• The epsilon symbol $\epsilon_{ijk}$ is in invariant tensor , $$ \epsilon_{ijk}G_{ii'}G_{jj'}G_{kk'} = \epsilon_{i'j'k'} $$ for $G\in {\rm SL}(3, {\mathbb C})$ , and any of its subgroups. One of these [stackexchange, 7 points]
• In some references, e.g in this Notes , identities involving the traces of matrices \begin{align} (D^a)^{bc}=d^{abc} \ , \quad (F^a)^{bc}=-if^{abc} \end{align} are given. For instance (eq.66) \begin{a [stackexchange, 1 points]

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