A note on non-simply connected rational homotopy models

In their generalization of the rational homotopy theory to non-simply connected spaces, G\'omez-Tato--Halperin--Tanr\'e adopted local systems of commutative differential graded algebras (CDGA's) as algebraic models. As another non-simply connected rational model, a $\pi_1$-equivariant CDGA with a semi-simple action was introduced by Hain and studied by Katzarkov-Pantev-To\"en and Pridham. These models have their own advantages and direct comparison might be important. In this note, we present an example of local system of CDGA's which is a non-nilpotent version of an example given by G\'omez-Tato et al., and compute the corresponding $\pi_1$-equivariant CDGA. We also see how some homotopy invariants such as homotopy groups and twisted cohomology are algebraically recovered from the equivariant CDGA with this example. We also describe a general procedure to obtain the corresponding equivariant CDGA model from a local system model in the case of $\pi_1=\mathbb{Z}\times \mathbb{Z}$.