The dual of a certain left quantum group

摘要

The second author and his collaborators, Lauve and Rodriguez-Romo (see [A class of left quantum groups modeled after SLq(n), J. Pure Appl. Algebra 208(3) (2007) 797-803; A left quantum group, J. Algebra 286 (2005) 154-160), have constructed left Hopf algebras which are not Hopf algebras modeled after SLq(n). In particular, they constructed the left quantum group <(SLq(2))over tilde>, along with an epimorphism to the quantum group SLq(2), the latter being a Hopf algebra. The current work began as a search for a left Hopf algebra (which is not a Hopf algebra) containing the quantum group U-q(sl(2)), the latter being a Hopf algebra. The natural choice was to look for the dual of <(SLq(2))over tilde>. We show that the Hopf dual of <(SLq(2))over tilde> is equal to the Hopf dual of SLq(2), which is of independent interest. Thus the Hopf dual of <(SLq(2))over tilde> is a Hopf algebra. The original search of a left Hopf algebra which is not a Hopf algebra, larger than U-q(sl(2)), is still open.