there exist bialgebras with a left antipode but no right antipode(j.a.green,w.d.nichols,e.j.taft,j.algebra 65,399-411). we try toconstruct such a left hopf algebra in the framework of quantum groups.we start with $3$ of the $6$ relations defining quantum $gl(2)$,plus inverting the quantum determinant. in asking that the left antipode, with itsstandard action on the $4$ generators, be an algebra antiendomorphism, weare forced to add new relations. the process stops at a hopf algebra( two-sided) which seems to be new. it has the unusual feature that itremains non-commutative when $q=1$. recently, we have dropped thecondition that the left antipode be an algebra antiendomorphism, but try to make it reverse the product only on irreducible words in thegenerators( there is a birkhoff-witt type basis). this almost works,but causes trouble on one nasty irreducible word. we hope to overcome this. ( joint work with suemi rodriguez-romo)