Papyrus 116

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)

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

$[J^{a},J^{b}]=i\epsilon ^{abc}J^{c}\,$

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 $\delta _{ab}$).

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

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

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

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:

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

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.