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In group theory, a discipline within mathematics, the structure constants of a Lie group determine the commutation relations between its generators in the associated Lie algebra.

Definition

Given a set of generators $T^i$, the structure constants $f^{abc}$express the Lie brackets of pairs of generators as linear combinations of generators from the set, i.e.

$[T^a,T^b]=f^{abc}{T}^c$.

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 $X,Y$ of the Lie algebra, the structure of the Lie group near the identity element is given by $\exp (X)\exp (Y)\approx \exp (X+Y+\frac{1}{2}[X,Y])$. This expression is made exact by the Baker–Campbell–Hausdorff formula.

Examples

SU(2)

The generators of the group SU(2) satisfy the commutation relations (where ${ϵ}^{abc}$ is the Levi-Civita symbol):

$[J^a,J^b]=i{ϵ}^{abc}{J}^c$

In this case, $f^{abc}=i{ϵ}^{abc}$, and the distinction between upper and lower indexes doesn't matter (the metric is the Kronecker delta ${\delta }_{ab}$).

SU(3)

A less trivial example is given by SU(3):

Its generators, T, in the defining representation, are:

$T^a=\frac{{\lambda }^a}{2}.$

where $\lambda$, the Gell-Mann matrices, are the SU(3) analog of the Pauli matrices for SU(2):

${\lambda }^1=\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right)$	${\lambda }^2=\left(\begin{array}{ccc}0& -i& 0\\ i& 0& 0\\ 0& 0& 0\end{array}\right)$	${\lambda }^3=\left(\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 0\end{array}\right)$
${\lambda }^4=\left(\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 1& 0& 0\end{array}\right)$	${\lambda }^5=\left(\begin{array}{ccc}0& 0& -i\\ 0& 0& 0\\ i& 0& 0\end{array}\right)$	${\lambda }^6=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right)$
${\lambda }^7=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& -i\\ 0& i& 0\end{array}\right)$	${\lambda }^8=\frac{1}{\sqrt{3}}\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -2\end{array}\right).$

These obey the relations

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

$\{T^a,T^b\}=\frac{1}{3}{\delta }^{ab}+d^{abc}{T}^c.$

The structure constants are given by:

$f^{123}=1$

$f^{147}=-f^{156}=f^{246}=f^{257}=f^{345}=-f^{367}=\frac{1}{2}$

$f^{458}=f^{678}=\frac{\sqrt{3}}{2},$

and all other $f^{abc}$ not related to these by permutation are zero.

The d take the values:

$d^{118}=d^{228}=d^{338}=-d^{888}=\frac{1}{\sqrt{3}}$

$d^{448}=d^{558}=d^{668}=d^{778}=-\frac{1}{2\sqrt{3}}$

$d^{146}=d^{157}=-d^{247}=d^{256}=d^{344}=d^{355}=-d^{366}=-d^{377}=\frac{1}{2}.$

Hall polynomials

The Hall polynomials are the structure constants of the Hall algebra.

Applications

• 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 $G_{\mu \nu }^a$ represents the gauge invariant gluon field strength tensor, analogous to the electromagnetic field strength tensor, F^μν, in quantum electrodynamics. It is given by:

$G_{\mu \nu }^a={\partial }_{\mu }{\mathcal{𝒜}}_{\nu }^a-{\partial }_{\nu }{\mathcal{𝒜}}_{\mu }^a+g{f}^{abc}{\mathcal{𝒜}}_{\mu }^b{\mathcal{𝒜}}_{\nu }^c,$

where f_abc are the structure constants of SU(3). Note that the rules to move-up or pull-down the a, b, or c indexes are trivial, (+,... +), so that f^abc = f_abc = fTemplate:Su whereas for the μ or ν indexes one has the non-trivial relativistic rules, corresponding e.g. to the metric signature (+ − − −).

References

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• Weinberg, Steven, The Quantum Theory of Fields, Volume 1: Foundations, Cambridge University Press, Cambridge, (1995). ISBN 0-521-55001-7.