Twisted noncommutative equivariant cohomology: Weil and Cartan models

We propose Weil and Cartan models for the equivariant cohomology of noncommutative spaces which carry a covariant action of Drinfel’d twisted symmetries. The construction is suggested by the noncommutative Weil algebra of Alekseev and Meinrenken (2000) [5]; we show how to implement a Drinfel’d twist of their models in order to take into account the noncommutativity of the spaces we are acting on. We also provide basic examples and properties of the twisted noncommutative equivariant cohomology.