  Lie algebroid Yang-Mills theories are a generalization of Yang-Mills gauge
theories, replacing the structural Lie algebra by a Lie algebroid E. In this
note we relax the conditions on the fiber metric of E 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 in part explicit so as to be accessible also to a
less mathematically oriented audience.
