Lie Algebroid Yang–Mills with matter fields

Lie Algebroid Yang–Mills theories are a generalization of Yang–Mills gauge theories, replacing the structural Lie algebra by a Lie Algebroid EE. In this note we relax the conditions on the fiber metric of EE for gauge invariance of the action functional. Coupling to scalar fields requires possibly nonlinear representations of Lie Algebroids. In all cases, gauge invariance is seen to lead to a condition of covariant constancy on the respective fiber metric in question with respect to an appropriate Lie Algebroid connection.The presentation is kept explicit in part so as to be also accessible to a less mathematically oriented audience.