Let be a bialgebra, possibly noncommutative, over a field and an -matrix over . is a matrix bialgebra with basis if
the set of entries of generates and
the comultiplication and counit satisfy the matrix equations (i.e. in components ) and (reading in components ).
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 on generators has a unique coalgebra structure making it a matrix bialgebra with basis . We call it the free matrix bialgebra of rank . Every bialgebra quotient of that bialgebra is a matrix bialgebra.
A matrix Hopf algebra with basis is often not a matrix bialgebra with basis : e.g. the commutative coordinate ring of is not a matrix bialgebra with respect to the obvious basis ; 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 is a matrix bialgebra and a matrix Hopf algebra with the same standard basis .