Papyrus 116: Difference between revisions

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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 <math>T^i</math>, the '''[[structure constants]]''' <math>f^{abc}</math>express the Lie brackets of pairs of generators as linear combinations of generators from the set, i.e.


:<math>[T^a, T^b] = f^{abc} T^c</math>.  
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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 <math>X, Y</math> of the Lie algebra, the structure of the Lie group near the identity element is given by <math>\exp(X)\exp(Y) \approx \exp(X + Y + \tfrac{1}{2}[X,Y])</math>. 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 <math>\epsilon^{abc}</math> is the [[Levi-Civita symbol]]):
 
: <math>[J^a, J^b] = i \epsilon^{abc} J^c \, </math>
 
In this case, <math>f^{abc} = i\epsilon^{abc}</math>, and the distinction between upper and lower indexes doesn't matter (the metric is the [[Kronecker delta]] <math>\delta_{ab}</math>).
 
=== SU(3) ===
 
A less trivial example is given by SU(3):
 
Its generators, ''T'', in the defining representation, are:
:<math>T^a = \frac{\lambda^a }{2}.\,</math>
where <math>\lambda \,</math>, the [[Gell-Mann matrices]], are the SU(3) analog of the [[Pauli matrices]] for SU(2):
 
:{| border="0" cellpadding="5" cellspacing="0"
|<math>\lambda^1 = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}</math>
|<math>\lambda^2 = \begin{pmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}</math>
|<math>\lambda^3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{pmatrix}</math>
|-
|<math>\lambda^4 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}</math>
|<math>\lambda^5 = \begin{pmatrix} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{pmatrix}</math>
|<math>\lambda^6 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}</math>
|-
|<math>\lambda^7 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix}</math>
|<math>\lambda^8 = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end{pmatrix}.</math>
|
|}
 
These obey the relations
:<math>\left[T^a, T^b \right] = i f^{abc} T^c \,</math>
:<math> \{T^a, T^b\} = \frac{1}{3}\delta^{ab} + d^{abc} T^c. \,</math>
The structure constants are given by:
:<math>f^{123} = 1 \,</math>
:<math>f^{147} = -f^{156} = f^{246} = f^{257} = f^{345} = -f^{367} = \frac{1}{2} \,</math>
:<math>f^{458} = f^{678} = \frac{\sqrt{3}}{2}, \,</math>
and all other <math>f^{abc}</math> not related to these by permutation are zero.
 
The ''d'' take the values:
:<math>d^{118} = d^{228} = d^{338} = -d^{888} = \frac{1}{\sqrt{3}} \,</math>
:<math>d^{448} = d^{558} = d^{668} = d^{778} = -\frac{1}{2\sqrt{3}} \,</math>
:<math>d^{146} = d^{157} = -d^{247} = d^{256} = d^{344} = d^{355} = -d^{366} = -d^{377} = \frac{1}{2}. \,</math>
 
===Hall polynomials===
The Hall polynomials are the structure constants of the [[Hall algebra]].
 
==Applications==
 
*A [[nilmanifold|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.<ref>Raghunathan, Chapter II, ''Discrete Subgroups of Lie Groups'', M. S. Raghunathan</ref>
*In [[quantum chromodynamics]], the symbol <math>G^a_{\mu \nu} \,</math> represents the gauge invariant [[gluon field strength tensor]], analogous to the [[electromagnetic tensor|electromagnetic field strength tensor]], ''F''<sup>μν</sup>, in [[quantum electrodynamics]]. It is given by:<ref>{{cite article|title=The field strength correlator from QCD sum rules
|author=M. Eidemüller, H.G. Dosch, M. Jamin
|year=1999
|publisher=
|location=Heidelberg, Germany
|journal=Nucl.Phys.Proc.Suppl.86:421-425,2000
|arxiv=hep-ph/9908318
|url=http://arxiv.org/pdf/hep-ph/9908318v1.pdf}}</ref>
 
::<math>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 \,,</math>
 
:where ''f<sub>abc</sub>'' 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<sup>abc</sup>'' = ''f<sub>abc</sub>'' = ''f''{{su|b=''bc''|p=''a''}} whereas for the ''μ'' or ''ν'' indexes one has the non-trivial ''relativistic'' rules, corresponding e.g. to the [[metric signature]] (+ − − −).  
 
== References ==
{{reflist}}
* [[Steven Weinberg|Weinberg, Steven]], ''The Quantum Theory of Fields, Volume 1: Foundations'', Cambridge University Press, Cambridge, (1995). ISBN 0-521-55001-7.
 
[[Category:Lie algebras]]

Latest revision as of 19:49, 18 August 2014


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