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ON THE QUASITRIANGULARITY OF Uq(sln)′

Published online by Cambridge University Press:  01 February 1998

SHLOMO GELAKI  and
SARA WESTREICH
SHLOMO GELAKI
Affiliation:
Department of Mathematics and Computer Science, Ben Gurion University of the Negev, Beer Sheva, Israel
SARA WESTREICH
Affiliation:
Interdisciplinary Department of the Social Science, Bar-Ilan University, Ramat-Gan, Israel
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Abstract

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].

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 57 , Issue 1 , February 1998 , pp. 105 - 125
Copyright
The London Mathematical Society 1998

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