On the Quasitriangularity of U q (sl n )′

The quantum group U q (sl n ) introduced by Drinfel'd [ 2 ] and Jimbo [ 5 ] is a Hopf algebra which is naturally paired with O q (SL n ), the coordinate ring of quantum SL n . When q is not a root of unity, the finite dimensional representation theory of U q (sl n ) is essentially the same as that of U (sl n ). Furthermore, it is known that U q (sl n ) is essentially a quasitriangular Hopf algebra [ 2 ]. When q is a root of unity the situation changes dramatically, and the representation theory of U (sl n ) is no longer effective. Moreover, U q (sl n ) is not quasitriangular. In this case one can consider the quotient Hopf algebra U q (sl n )′, introduced by Lusztig [ 7 ], which is a finite dimensional Hopf algebra with a nice representation theory. It is well known that U q (sl 2 )′ is quasitriangular. Finite dimensional quasitriangular Hopf algebras are important for the study of knot invariants [ 11 , 12 ]. Thus, a natural question is: when is U q (sl n )′ quasitriangular? The somewhat unexpected answer is given in Theorem 3.7: it depends sharply on the greatest common divisor of n and the order of q 1/2 . For these Hopf algebras we classify all the possible R ‐matrices and give necessary and sufficient conditions for them to be minimal quasitriangular. These conditions depend again on n and the order of q 1/2 . In the process we describe the groups of Hopf automorphisms of U q (sl n )′ and O q (SL n )′, where O q (SL n )′ is a finite dimensional quotient Hopf algebra of Q q (SL n ) proved to be the dual of U q (sl n )′ in [ 15 ].