The Cartan homotopy formula was introduced for differentiable manifolds relating a vector field and its Lie derivative. Reinhart gave an algebraic analogue over a commutative algebra for $D$ a derivation. Getzler formulated the same for $A_\infty$ algebras and for $D$ an arbitrary cocycle.

We talk about the equivariant Cartan homotopy formula for cross product $DG$-algebras.