Braided coproduct, antipode and adjoint action for $U_q(sl_2)$

Pavle Pandžić and Petr Somberg

Address:
Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia
Mathematical Institute MFF UK, Sokolovská 83, 18000 Praha 8 - Karlín, Czech Republic

E-mail: pandzic@math.hr     somberg@karlin.mff.cuni.cz

Abstract: Motivated by our attempts to construct an analogue of the Dirac operator in the setting of $U_q(\mathfrak{sl}_n)$, we write down explicitly the braided coproduct, antipode, and adjoint action for quantum algebra $U_q(\mathfrak{sl}_2)$. The braided adjoint action is seen to coincide with the ordinary quantum adjoint action, which also follows from the general results of S. Majid.

AMSclassification: primary 16T20; secondary 20G42.

Keywords: quantum group, quantum \mathfrak{sl}_2, quantum adjoint action, tensor categories, braided tensor product, braided adjoint action.

DOI: 10.5817/AM2024-5-365