We demonstrate that the exterior algebra of an Atiyah Lie algebroid generalizes the familiar notions of the physicist's BRST complex. To reach this conclusion, we develop a general picture of Lie algebroid isomorphisms as commutative diagrams between algebroids preserving the geometric structure encoded in their brackets. Such a diagram defines a Lie algebroid morphism when the two algebroids possess gauge equivalent connections. This indicates that the set of Lie algebroid isomorphisms should be regarded as equivalent to the set of diffeomorphisms and gauge transformations. Moreover, a Lie algebroid isomorphism preserving the exterior algebra ensures that isomorphic algebroids are cohomologically equivalent. The Atiyah Lie algebroids derived from principal bundles with common base manifolds and structure groups may therefore be divided into equivalence classes of isomorphic algebroids. Each equivalence class possesses a trivialized Lie algebroid as a local representative, and the exterior algebra of the trivialized algebroid gives rise to the BRST complex. We then illustrate the usefulness of Lie algebroid cohomology in computing quantum anomalies, including applications to the chiral and Lorentz-Weyl (LW) anomalies. In particular, we pay close attention to the fact that the geometric intuition afforded by the Lie algebroid (which was absent in the naive BRST complex) provides hints of a deeper picture that simultaneously geometrizes the consistent and covariant forms of the anomaly. In the algebroid construction, the difference between the consistent and covariant anomalies is simply a different choice of basis.
