U_q(sl(m+1))-Module Algebra Structures on the Coordinate Algebra of a Quantum Vector Space

Steven Duplij¹ ; Yanyong Hong² ; Fang Li²

¹Mathematisches Institut, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany
²Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China

Journal of Lie Theory, Tome 25 (2015) no. 2, pp. 327-361

Résumé

We study the module-algebra structures of $U_q(sl(m+1))= {\cal H}(e_i,f_i,k_i^{\pm1})_{1\leq i\leq m}$ on the coordinate algebra of quantum vector spaces are studied. We denote the coordinate algebra of quantum $n$-dimensional vector space by $A_q(n)$. As our main result, first, we give a complete classification of module-algebra structures of $U_q(sl(m+1))$ on $A_q(3)$ when $k_i\in {\rm Aut L}(A_q(3))$ as actions on $A_q(3)$ for $i=1,\cdots, m$ and $m\geq 2$ and with the same method, on $A_q(2)$, all module-algebra structures of $U_q(sl(m+1))$ are characterized. Lastly, the module-algebra structures of $U_q(sl(m+1))$ on $A_q(n)$ are obtained for any $n\geq 4$.

Détail

Classification : 81R50, 16T20, 17B37, 20G42, 16S40
Mots-clés : Quantum enveloping algebra, coordinate algebra of quantum vector space, Hopf action, module algebra, weight