nLab
Hopf envelope

For any family {C i} iI\{ C_i \}_{i\in I} of coalgebras, T( iC i)= iT(C i)T(\coprod_i C_i) = \coprod_i T(C_i) 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 C i+1=C i rmcopC_{i+1} = C_i^{\rm cop}, the left hand side specializes to an intermediate stage in building Takuechi’s free Hopf algebra H(C)H(C) on the coalgebra CC.

Manin generalized the RHS. He replaces T(C i)T(C_i) with any bialgebra B iB_i with B i+1=B i rmcop,opB_{i+1} = B_i^{\rm cop, op}. Notice that the algebra structure is also opposite between even and odd cases (a superfluous/iunvisible condition in the case of the tensor algebra T(C i)T(C_i) appearing in Takeuchi's construction). Let = iB i\mathcal{B} = \coprod_i B_i and S:S : \mathcal{B} \rightarrow \mathcal{B} be again defined by a shift in index by +1+1. Then the 2-sided ideal I SI_S \subset \mathcal{B} generated by relations b (1)S(b (2))ϵ(b)1\sum b_{(1)} S(b_{(2)}) - \epsilon(b) 1 and S(b (1))b (2)ϵ(b)1\sum S(b_{(1)}) b_{(2)} - \epsilon(b) 1, for all bB ib \in B_i, is SS-stable ideal and the quotient H(B)=/I SH(B) = \mathcal{B}/I_S is a Hopf algebra, the Hopf envelope of the bialgebra BB.

It satisfies the following universal property: for any Hopf algebra HH' and a bialgebra map ϕ:B 0H\phi : B_0 \rightarrow H' there is a unique Hopf algebra map H(ϕ):H(B)HH(\phi) : H(B) \rightarrow H' such that H(ϕ)i=ϕH(\phi) \circ i = \phi where i:B 0H(B)i : B_0 \rightarrow H(B) is the composition of the inclusion into \mathcal{B} and the canonical projection H(B)\mathcal{B} \rightarrow H(B).

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 t j it^i_j whose underlying bialgebra is the free bialgebra on n 2n^2 generators t j it^i_j is sometimes called the free matrix Hopf algebra, cf. section 13 of

for more details.