On minimal quasitriangular pointed Hopf~ algebras

J. Math. Sci. Univ. Tokyo
Vol. 16 (2009), No. 4, Page 545--576.

Masuoka, Akira
On minimal quasitriangular pointed Hopf~ algebras
[Full Article (PDF)] [MathSciNet Review (HTML)] [MathSciNet Review (PDF)]

The quantized enveloping algebra $U_q$ is constructed as a quotient of the generalized quantum double $ U^{\leq 0}_q \cmdbicross_{\tau} U^{\geq 0}_q $ associated to a natural skew pairing $ \tau : U^{\leq 0}_q \otimes U^{\geq 0}_q \rightarrow k $. This double is generalized by \begin{equation*} \cmdcalD = ( \cmdfrakB ( V ) \cmddotrtimes F ) \cmdbicross_{\tau} ( \cmdfrakB ( W ) \cmddotrtimes G ), \end{equation*} where $F$, $G$ are abelian groups, $ V \in {}^F_F \cmdcalYD $, $ W \in {}^G_G \cmdcalYD $ are Yetter-Drinfeld modules and $ \cmdfrakB ( V ) $, $ \cmdfrakB ( W ) $ are their Nichols algebras. We prove some results on Hopf ideals of $ \cmdcalD $, including a characterization of what we call thin Hopf ideals. As an application we give an explicit description of those minimal quasitriangular pointed Hopf algebras in characteristic zero which are generated by skew primitives.

Keywords: Hopf algebra, Nichols algebra, generalized quantum double, quasitriangular

Mathematics Subject Classification (2000): 16W30, 17B37
Mathematical Reviews Number: MR2650520

Received: 2009-12-18