by Westreich, Sara

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 $G(U_{\!Q})\cong G(U_{Q’}).$ In the last part we show that U

_Q
is always a quotient of a double crossproduct.

DOI: 10.1007/s10468-007-9079-9
Online Date: 6/28/2007
Print publication date: 3/1/2008
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