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تسجيل مستخدم جديدTriangular decomposition of right coideal subalgebras
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تأليف
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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