  Bloch and Kriz construct an abelian category of mixed Tate motives as the
category of comodules over a Hopf algebra obtained by the bar construction of
the DGA of cycle complexes. In this paper we generalize their construction to
give the definition of a category of mixed elliptic motives, i.e. a Tannakian
category of mixed motives generated by an elliptic curve. We introduce the
notion of a relative DGA over a reductive group. Then the category of mixed
elliptic motives is defined as the category of comodules over the relative bar
construction of a certain DGA $A_{EM}$ which is constructed from cycle
complexes. The elliptic polylogarithm of Beilinson-Levin gives an interesting
object in this category.
