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Register a new userTriangular decomposition of right coideal subalgebras
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V.K. Kharchenko
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Let $mathfrak g$ be a Kac-Moody algebra. We show that every homogeneous right coideal subalgebra $U$ of the multiparameter version of the quantized universal enveloping algebra $U_q(mathfrak{g}),$ $q^m eq 1$ containing all group-like elements has a triangular decomposition $U=U^-otimes_{{bf k}[F]} {bf k}[H] otimes_{{bf k}[G]} U^+$, where $U^-$ and $ U^+$ are right coideal subalgebras of negative and positive quantum Borel subalgebras. However if $ U_1$ and $ U_2$ are arbitrary right coideal subalgebras of respectively positive and negative quantum Borel subalgebras, then the triangular composition $ U_2otimes_{{bf k}[F]} {bf k}[H]otimes_{{bf k}[G]} U_1$ is a right coideal but not necessary a subalgebra. Using a recent combinatorial classification of right coideal subalgebras of the quantum Borel algebra $U_q^+(mathfrak{so}_{2n+1}),$ we find a necessary condition for the triangular composition to be a right coideal subalgebra of $U_q(mathfrak{so}_{2n+1}).$ If $q$ has a finite multiplicative order $t>4,$ similar results remain valid for homogeneous right coideal subalgebras of the multiparameter version of the small Lusztig quantum groups $u_q({frak g}),$ $u_q(frak{so}_{2n+1}).$
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Let $H$ be a character Hopf algebra. Every right coideal subalgebra that contains the coradical has a PBW-basis which can be extended up to a PBW-basis of $H.$
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 $Ucap G=Sigma .$ 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].
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