Hopf algebras of type A_n, 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.