Triangular decomposition of right coideal subalgebras

Abstract

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 $U_q(\mathfrak{g}),$ $q^m\neq 1$ 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_2\otimes_{{\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}).

Gröbner–Shirshov bases and their calculation

“…Lusztig's theorem [154] provides linear canonical bases of these algebras. Different approaches were developed by Ringel [191,192], Green [110], and Kharchenko [131][132][133][134][135]. GS bases of quantum enveloping algebras are unknown except for the case A n , see [55,86,195,220] [151]).…”

Homogeneous right coideal subalgebras of quantized enveloping algebras

“…The triangular decomposition of U descends to the level of homogeneous right coideal subalgebras of U . This result is in principle contained in [Let02, Section 4] and was also proved in [Kha10]. Here we recall the argument for the convenience of the reader.…”

Gröbner-Shirshov bases and their calculation

“…Recently the CD-lemmas mentioned above and other combinatorial methods yielded many applications: for groups of Novikov-Boone type [119,120,121] (see also [16,17,77,118], for Coxeter groups [58,151], for center-bymetabelian Lie algebras [214], for free metanilpotent Lie algebras, Lie algebras and associative algebras [112,168,215,216], for Poisson algebras [159], for quantum Lie algebras and related problems [132,135], for PBW-bases [131,134,158], for extensions of groups and associative algebras [73,74], for (color) Lie (p)-superalgebras [9,48,91,92,105,106,107,169,227,228], for Hecke algebras and Specht modules [125], for representations of Ariki-Koike algebras [126], for the linear algebraic approach to GS bases [127], for HNN groups [87], for certain one-relator groups [88], for embeddings of algebras [39,83], for free partially commutative Lie algebras [84,181], for quantum groups of type D n , E 6 , and G 2…”

“…A. Green [110], and V. K. Kharchenko [131,132,133,134,135]. GS bases of quantum enveloping algebras are unknown except for the case A n , see [55,86,195,220].…”