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] = 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] = 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 $ℝ 3$ , 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

↑ Raghunathan, Chapter II, Discrete Subgroups of Lie Groups, M. S. Raghunathan
↑ M. Eidemüller, H.G. Dosch, M. Jamin (1999). "The field strength correlator from QCD sum rules". 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.