, Volume 61, Issue 1, pp 41-50

Some Quantum-like Hopf Algebras which Remain Noncommutative when q = 1

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Abstract

Starting with only three of the six relations defining the standard (Manin) GL q (2), we try to construct a quantum group. The antipode condition requires some new relations, but the process stops at a Hopf algebra with a Birkhoff–Witt basis of irreducible monomials. The quantum determinant is group-like but not central, even when q = 1. So, the two Hopf algebras constructed in this way are not isomorphic to the Manin GL q (2), all of whose group-like elements are central. Analogous constructions can be made starting with the Dipper–Donkin version of GL q (2), but these turn out to be included in the two classes of Hopf algebras described above.