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

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

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 $t^i_j$ whose underlying bialgebra is the free bialgebra on $n^2$ generators $t^i_j$ 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.