Structure constants

From Wikipedia, the free encyclopedia

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.^[1]
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:^[2]

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

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 = 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.

This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.