nLab
matrix Hopf algebra

Let $B$ be a bialgebra, possibly noncommutative, over a field $k$ and $G=(g_j^i{)}_{j=1,\dots ,n}^{i=1,\dots ,n}$ an $n\times n$-matrix over $B$. $B$ is a matrix bialgebra with basis $G$ if

• the set of entries of $G$ generates $B$ and

• the comultiplication $\Delta$ and counit $ϵ$ satisfy the matrix equations $\Delta G=G\otimes G$ (i.e. in components $\Delta g_j^i={\sum }_{k=1}^n{g}_k^i\otimes g_j^k$) and $ϵG=1$ (reading in components $ϵ(g_j^i)={\delta }_j^i$).

According to a result of Redford every finite-dimensional Hopf algebra over a field is a matrix Hopf algebra with respect to some basis.

The free (noncommutative) associative algebra $F$ on $n^2$ generators $f_j^i$ has a unique coalgebra structure making it a matrix bialgebra with basis $(f_j^i{)}_{j=1,\dots ,n}^{i=1,\dots ,n}$. We call it the free matrix bialgebra of rank $n^2$. Every bialgebra quotient of that bialgebra is a matrix bialgebra.

A matrix Hopf algebra $\mathscr{H}$ with basis $T=(t_j^i)$ is a Hopf algebra which possess a matrix subbialgebra $B$ with basis $T$ such that the map $H({\mathrm{id}}_B):H(B)\to \mathscr{H}$ is onto (where $H(B)$ denotes the Hopf envelope of $B$ and $H$ is understood as a functor).

A matrix Hopf algebra $\mathscr{H}$ with basis $T$ is often not a matrix bialgebra with basis $T$: e.g. the commutative coordinate ring of $\mathrm{GL}(n,k)$ is not a matrix bialgebra with respect to the obvious basis $T$; in this example this can be repaired by enlarging the basis by one group-like element: the inverse of the determinant. On the other hand, the coordinate algebra of the special linear group $O(\mathrm{SL}(n,k))$ is a matrix bialgebra and a matrix Hopf algebra with the same standard basis $T$.