The quantum group Uq(SLn)
introduced by Drinfel'd [2] and Jimbo [5]
is a Hopf
algebra which is naturally paired with Oq(sln),
the coordinate ring of quantum SLn.
When q is not a root of unity, the finite dimensional representation
theory of
Uq(sln)
is essentially the same as that of U(sln).
Furthermore, it is known that Uq(sln)
is
essentially a quasitriangular Hopf algebra [2].
When q is a root of unity the situation
changes dramatically, and the representation theory of
U(sln) is no longer effective.
Moreover, Uq(sln) is
not quasitriangular. In this case one can consider the quotient
Hopf algebra Uq(sln)′,
introduced by Lusztig [7], which is a finite dimensional
Hopf
algebra with a nice representation theory. It is well known that
Uq(sl2)′ is
quasitriangular. Finite dimensional quasitriangular Hopf algebras are important
for
the study of knot invariants [11, 12].
Thus, a natural question is: when is
Uq(sln)′
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
q1/2. 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 q1/2.
In the process we describe the groups of Hopf automorphisms
of Uq(sln)′ and
Oq(SLn)′, where
Oq(SLn)′ is a
finite dimensional quotient Hopf algebra
of Qq(SLn) proved to be
the
dual of Uq(sln)′
in
[15].