Right coideal subalgebras in $U_q(\frak{sl}_{n+1}).$

Abstract

We offer a complete classification of right coideal subalgebras which contain all group-like elements for the multiparameter version of the quantum group $U_q(\mathfrak{sl}_{n+1})$ provided that the main parameter $q$ is not a root of 1. As a consequence, we determine that for each subgroup $\Sigma $ of the group $G$ of all group-like elements the quantum Borel subalgebra $U_q^+ (\mathfrak{sl}_{n+1})$ containes $(n+1)!$ different homogeneous right coideal subalgebras $U$ such that $U\cap G=\Sigma .$ If $q$ has a finite multiplicative order $t>2,$ the classification remains valid for homogeneous right coideal subalgebras of the multiparameter version of the Lusztig quantum group $u_q (\frak{sl}_{n+1}).$ In the paper we consider the quantifications of Kac-Moody algebras as character Hopf algebras [V.K. Kharchenko, A combinatorial approach to the quantifications of Lie algebras, Pacific J. Math., 203(1)(2002), 191- 233].