For any family of coalgebras, where the coproduct on the right-hand side is the coproduct in the category of bialgebras, i.e. the free product of algebras with natural induced coalgebra structure. If the index set consists of nonnegative integers and , the left hand side specializes to an intermediate stage in building Takuechi’s free Hopf algebra on the coalgebra .
Manin generalized the RHS. He replaces with any bialgebra with . Notice that the algebra structure is also opposite between even and odd cases (a superfluous/iunvisible condition in the case of the tensor algebra appearing in Takeuchi's construction). Let and be again defined by a shift in index by . Then the 2-sided ideal generated by relations and , for all , is -stable ideal and the quotient is a Hopf algebra, the Hopf envelope of the bialgebra .
It satisfies the following universal property: for any Hopf algebra and a bialgebra map there is a unique Hopf algebra map such that where is the composition of the inclusion into and the canonical projection .
Manin has introduced this construction in
and applied it mainly to matrix Hopf algebras (e.g. quantum linear groups). The Hopf envelope of the matrix Hopf algebra with basis whose underlying bialgebra is the free bialgebra on generators is sometimes called the free matrix Hopf algebra, cf. section 13 of
for more details.