Hopf Algebras of Type An, Twistings and the FRT-construction

We study pointed Hopf algebras of the form U ( R Q ), (Faddeev et al., Quantization of Lie groups and Lie algebras. Algebraic Analysis , vol. I, Academic, Boston, MA, pp. 129–139, 1988 ; Faddeev et al., Quantum groups. Braid group, knot theory and statistical mechanics. Adv. Ser. Math. Phys. , vol. 9, World Science, Teaneck, NJ, pp. 97–110, 1989 ; Larson and Towber, Commun. Algebra 19(12):3295–3345, 1991 ), where R Q is the Yang–Baxter operator associated with the multiparameter deformation of GL n supplied in Artin et al. ( Commun. Pure Appl. Math. 44:8–9, 879–895, 1991 ) and Sudbery ( J. Phys. A , 23(15):697–704, 1990 ). We show that U ( R Q ) is of type A n in the sense of Andruskiewitsch and Schneider ( Adv. Math. 154:1–45, 2000 ; Pointed Hopf algebras. Recent developments in Hopf Algebras Theory, MSRI Series , Cambridge University Press, Cambridge, 2002 ). We consider the non-negative part of U ( R Q ) and show that for two sets of parameters, the corresponding Hopf sub-algebras can be obtained from each other by twisting the multiplication if and only if they possess the same groups of grouplike elements. We exhibit families of finite-dimensional Hopf algebras arising from U ( R Q ) with non-isomorphic groups of grouplike elements. We then discuss the case when the quantum determinant is central in A ( R Q ) and show that under some assumptions on the group of grouplike elements, two finite-dimensional Hopf algebras U ( R Q ), U ( R Q ′ ) can be obtained from each other by twisting the comultiplication if and only if In the last part we show that U Q is always a quotient of a double crossproduct.