Affine group dg-schemes and linear representations I - Basic theory and Tannakian reconstructions

We develop a basic theory of affine group dg-schemes, their Lie algebraic counterparts and linear representations. We prove Tannaka type reconstruction theorems that an affine group dg-scheme can be recovered from the dg-tensor category of its linear representations as well as from the rigid dg-tensor category of its finite dimensional linear representations along with the forgetful functors to the underlying dg-tensor category of cochain complexes.