Structure constants

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 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:

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

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