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We propose a generalized way to formally obtain the gauge invariance of thekinetic part of a field Lagrangian over which a gauge transformation ruled byan $SU(n)_{U} \otimes SU(m)_{V}$ coupling symmetry is applied. As anillustrative example, we employ such a formal construction for reproducing thestandard model Lagrangian. This generalized formulation is supposed tocontribute for initiating the study of gauge transformation applied togeneralized $SU(n)_{U} \otimes SU(m)_{V}$ symmetries as well as forcomplementing an introductory study of the standard model of elementaryparticles.

Obtaining the Gauge Invariant Kinetic Term for a SU(n)$_U$ x SU(m)$_V$ Lagrangian