Given a set of generators , the structure constants express the Lie brackets of pairs of generators as linear combinations of generators from the set, i.e.
The structure constants determine the Lie brackets of elements of the Lie algebra, and consequently nearly completely determine the group structure of the Lie group. For small elements of the Lie algebra, the structure of the Lie group near the identity element is given by . This expression is made exact by the Baker–Campbell–Hausdorff formula.
In this case, , and the distinction between upper and lower indexes doesn't matter (the metric is the Kronecker delta ).
A less trivial example is given by SU(3):
Its generators, T, in the defining representation, are:
These obey the relations
The structure constants are given by:
The d take the values:
The Hall polynomials are the structure constants of the Hall algebra.
- A lie group is abelian exactly when all structure constants are 0.
- A lie group is real exactly when its structure constants are real.
- A nilpotent Lie group admits a lattice if and only if its Lie algebra admits a basis with rational structure constants: this is Malcev's criterion. Not all nilpotent Lie groups admit lattices; for more details, see also Raghunathan.
- In quantum chromodynamics, the symbol represents the gauge invariant gluon field strength tensor, analogous to the electromagnetic field strength tensor, Fμν, in quantum electrodynamics. It is given by:
- where fabc are the structure constants of SU(3). Note that the rules to push-up or pull-down the a, b, or c indexes are trivial, (+,... +), so that fabc = fabc = fTemplate:Su whereas for the μ or ν indexes one has the non-trivial relativistic rules, corresponding e.g. to the metric signature (+ − − −).
- Raghunathan, Chapter II, Discrete Subgroups of Lie Groups, M. S. Raghunathan
- Template:Cite article
- Weinberg, Steven, The Quantum Theory of Fields, Volume 1: Foundations, Cambridge University Press, Cambridge, (1995). ISBN 0-521-55001-7.