Triangular decomposition of right coideal subalgebras

TL;DR

This paper proves that homogeneous right coideal subalgebras of multiparameter quantum groups have a triangular decomposition, and explores conditions under which their compositions form subalgebras, with implications for small Lusztig quantum groups.

Contribution

It establishes a triangular decomposition for homogeneous right coideal subalgebras of multiparameter quantum groups and analyzes when their compositions are subalgebras, extending to small Lusztig quantum groups.

Findings

Homogeneous right coideal subalgebras admit a triangular decomposition.

The triangular composition of certain subalgebras is generally not a subalgebra.

Necessary conditions are identified for the composition to be a right coideal subalgebra.

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 $u_q({\frak g}),$ $u_q(\frak{so}_{2n+1}).$