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

Content Provider	Springer Nature Link
Author	Westreich, Sara
Copyright Year	2007
Abstract	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.
Ending Page	82
Page Count	20
Starting Page	63
File Format	PDF
ISSN	1386923X
e-ISSN	15729079
Journal	Algebras and Representation Theory
Issue Number	1
Volume Number	11
Language	English
Publisher	Springer Netherlands
Publisher Date	2007-06-28
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Associative Rings and Algebras Hopf Algebras Quasitriangular Non-associative Rings and Algebras Twistings FRT-construction Biproducts Commutative Rings and Algebras Grouplike elements
Content Type	Text
Resource Type	Article
Subject	Mathematics

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