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 ${\displaystyle f^{abc}=i\epsilon ^{abc}}$, where ${\displaystyle \epsilon ^{abc}}$ 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 ${\displaystyle T^{i}}$, the structure constants ${\displaystyle f^{abc}}$express the Lie brackets of pairs of generators as linear combinations of generators from the set, i.e.

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

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 ${\displaystyle X,Y}$ of the Lie algebra, the structure of the Lie group near the identity element is given by ${\displaystyle \exp(X)\exp(Y)\approx \exp(X+Y+{\tfrac {1}{2}}[X,Y])}$. 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 ${\displaystyle \sigma _{i}}$. The generators of the group SU(2) satisfy the commutation relations (where ${\displaystyle \epsilon ^{abc}}$ is the Levi-Civita symbol):

${\displaystyle [\sigma _{a},\sigma _{b}]=i\epsilon ^{abc}\sigma _{c}\,}$

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

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:

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

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

These obey the relations

${\displaystyle \left[T^{a},T^{b}\right]=if^{abc}T^{c}\,}$
${\displaystyle \{T^{a},T^{b}\}={\frac {1}{3}}\delta ^{ab}+d^{abc}T^{c}.\,}$

The structure constants are given by:

${\displaystyle f^{123}=1\,}$
${\displaystyle f^{147}=-f^{156}=f^{246}=f^{257}=f^{345}=-f^{367}={\frac {1}{2}}\,}$
${\displaystyle f^{458}=f^{678}={\frac {\sqrt {3}}{2}},\,}$

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

The d take the values:

${\displaystyle d^{118}=d^{228}=d^{338}=-d^{888}={\frac {1}{\sqrt {3}}}\,}$
${\displaystyle d^{448}=d^{558}=d^{668}=d^{778}=-{\frac {1}{2{\sqrt {3}}}}\,}$
${\displaystyle d^{146}=d^{157}=-d^{247}=d^{256}=d^{344}=d^{355}=-d^{366}=-d^{377}={\frac {1}{2}}.\,}$

Hall polynomials

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

Applications

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

References

1. Raghunathan, Chapter II, Discrete Subgroups of Lie Groups, M. S. Raghunathan
2. Template:Cite article
• Weinberg, Steven, The Quantum Theory of Fields, Volume 1: Foundations, Cambridge University Press, Cambridge, (1995). ISBN 0-521-55001-7.