Structure constants

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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 , where 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 , the structure constants express the Lie brackets of pairs of generators as linear combinations of generators from the set, i.e.

.

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 of the Lie algebra, the structure of the Lie group near the identity element is given by . 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 . The generators of the group SU(2) satisfy the commutation relations (where is the Levi-Civita symbol):

In this case, , and the distinction between upper and lower indexes doesn't matter (the metric is the Kronecker delta ).

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:

where , the Gell-Mann matrices, are the SU(3) analog of the Pauli matrices for SU(2):

These obey the relations

The structure constants are given by:

and all other not related to these by permutation are zero.

The d take the values:

Hall polynomials

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

Applications

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.