Hopf Algebras of Type A n , Twistings and the FRT-construction

Abstract

We study pointed Hopf algebras of the form U(RQ), (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 RQ is the Yang–Baxter operator associated with the multiparameter deformation of GLn 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(RQ) is of type An 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(RQ) 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(RQ) with non-isomorphic groups of grouplike elements. We then discuss the case when the quantum determinant is central in A(RQ) and show that under some assumptions on the group of grouplike elements, two finite-dimensional Hopf algebras U(RQ), U(RQ′) can be obtained from each other by twisting the comultiplication if and only if \(G(U_{\!Q})\cong G(U_{Q'}).\) In the last part we show that UQ is always a quotient of a double crossproduct.