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Covariant Weil algebras

In this paper we introduce the classical and quantum covariant Weil algebras. Covariant Weil algebras are simultaneous generalizations of Weil algebras and family algebras. We will define differentials, Lie derivatives and contractions on them to make them curved-dg algebras. Moreover, the expression of curvatures will be given. It is hoped that covariant Weil algebras can be used in the construction of Mackey's analogue.