Structure constants

Using the cross product as a lie bracket, the algebra of 3-dimensional real vectors is a lie algebra isomorphic to the lie algebras of SU(2) and SO(3). In all three cases, the structure constants $f^{abc} = i\epsilon^{abc}$, where $\epsilon^{abc}$ is the completely antisymmetric tensor.

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.

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

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 $\exp(X)\exp(Y) \approx \exp(X + Y + \tfrac{1}{2}[X,Y])$. This expression is made exact by the Baker–Campbell–Hausdorff formula.

Examples

SU(2)

This algebra is three-dimensional, with generators given by the Pauli matrices $\sigma_i$. The generators of the group SU(2) satisfy the commutation relations (where $\epsilon^{abc}$ is the Levi-Civita symbol):

$[\sigma_a, \sigma_b] = 2 i \epsilon^{abc} \sigma_c \,$

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

This lie algebra is isomorphic to the lie algebra of SO(3),and also to the Clifford algebra of ℝ³, called the algebra of physical space.

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):

$\lambda^1 = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$	$\lambda^2 = \begin{pmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$	$\lambda^3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{pmatrix}$
$\lambda^4 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}$	$\lambda^5 = \begin{pmatrix} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{pmatrix}$	$\lambda^6 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}$
$\lambda^7 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix}$	$\lambda^8 = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end{pmatrix}.$

These obey the relations

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

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

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 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.^[1]
• In quantum chromodynamics, the symbol $G^a_{\mu \nu} \,$ represents the gauge invariant gluon field strength tensor, analogous to the electromagnetic field strength tensor, F^μν, in quantum electrodynamics. It is given by:^[2]

$G^a_{\mu \nu} = \partial_\mu \mathcal{A}^a_\nu - \partial_\nu \mathcal{A}^a_\mu + g f^{abc} \mathcal{A}^b_\mu \mathcal{A}^c_\nu \,,$

where f_abc 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 f^abc = f_abc = fa
bc whereas for the μ or ν indexes one has the non-trivial relativistic rules, corresponding e.g. to the metric signature (+ − − −).

References

1. Raghunathan, Chapter II, Discrete Subgroups of Lie Groups, M. S. Raghunathan
2. M. Eidemüller, H.G. Dosch, M. Jamin (1999). "The field strength correlator from QCD sum rules" (PDF). Nucl.Phys.Proc.Suppl.86:421-425,2000 (Heidelberg, Germany). arXiv:hep-ph/9908318.

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