nLab
Hopf envelope

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

Manin generalized the RHS. He replaces $T(C_i)$ with any bialgebra $B_i$ with $B_{i+1}=B_i^{rm\mathrm{cop},\mathrm{op}}$. Notice that the algebra structure is also opposite between even and odd cases (a superfluous/iunvisible condition in the case of the tensor algebra $T(C_i)$ appearing in Takeuchi's construction). Let $ℬ={\coprod }_i{B}_i$ and $S:ℬ\to ℬ$ be again defined by a shift in index by $+1$. Then the 2-sided ideal $I_S\subset ℬ$ generated by relations $\sum b_{(1)}S(b_{(2)})-ϵ(b)1$ and $\sum S(b_{(1)})b_{(2)}-ϵ(b)1$, for all $b\in B_i$, is $S$-stable ideal and the quotient $H(B)=ℬ/I_S$ is a Hopf algebra, the Hopf envelope of the bialgebra $B$.

It satisfies the following universal property: for any Hopf algebra $H\prime$ and a bialgebra map $\varphi :B_0\to H\prime$ there is a unique Hopf algebra map $H(\varphi ):H(B)\to H\prime$ such that $H(\varphi )\circ i=\varphi$ where $i:B_0\to H(B)$ is the composition of the inclusion into $ℬ$ and the canonical projection $ℬ\to H(B)$.

Manin has introduced this construction in

• Yu. I. Manin, Quantum groups and non-commutative geometry, CRM, Montreal 1988,

and applied it mainly to matrix Hopf algebras (e.g. quantum linear groups). The Hopf envelope of the matrix Hopf algebra with basis $t_j^i$ whose underlying bialgebra is the free bialgebra on $n^2$ generators $t_j^i$ is sometimes called the free matrix Hopf algebra, cf. section 13 of

• Z. Škoda, Localizations for construction of quantum coset spaces, math.QA/0301090, “Noncommutative geometry and Quantum groups”, W.Pusz, P.M. Hajac, eds. Banach Center Publications vol.61, pp. 265–298, Warszawa 2003.

for more details.