Non-abelian gauge transformation

In theoretical physics, a non-abelian gauge transformation means a gauge transformation taking values in some group G, the elements of which do not obey the commutative law when they are multiplied. By contrast, the original choice of gauge group in the physics of electromagnetism had been U(1), which is commutative.

For a non-abelian Lie group G, its elements do not commute, i.e. they in general do not satisfy

The quaternions marked the introduction of non-abelian structures in mathematics.

In particular, its generators ${\displaystyle t^{a}}$, which form a basis for the vector space of infinitesimal transformations (the Lie algebra), have a commutation rule:

The structure constants ${\displaystyle C^{abc}}$ quantify the lack of commutativity, and do not vanish. We can deduce that the structure constants are antisymmetric in the first two indices and real. The normalization is usually chosen (using the Kronecker delta) as

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