A left quantum group

Bialgebras with a left antipode but no right antipode were constructed in the early 1980s in [J.A. Green et al., Left Hopf algebras, J. Algebra 65 (1980) 399–411; W.D. Nichols, E.J. Taft, The left antipodes of a left Hopf algebra, in: Contemp. Math., vol. 13, Amer. Math. Soc., 1982, pp. 363–368]. Recently, in [S. Rodríguez-Romo, E.J. Taft, Some quantum-like Hopf algebras which remain noncommutative when q=1, Lett. Math. Phys. 61 (2002) 41–50] we tried to construct such a one-sided Hopf algebra within the framework of quantum groups, starting with roughly half the defining relations for quantum GL(2). Asking that the left antipode constructed be an algebra antimorphism led to some additional relations, but the result was a new (two-sided) Hopf algebra. Now we start with roughly half the relations for quantum SL(2) but ask that our left antipode constructed reverse order only on irreducible monomials in the generators. The result is a quantum group with a left antipode but no right antipode.