[Paper Review] Right coideal subalgebras in U^+_q(so_{2n+1})

V. K. Kharchenko|ArXiv.org|Aug 28, 2009

Algebraic structures and combinatorial models4 references5 citations

TL;DR

This paper provides a complete classification of right coideal subalgebras containing the coradical in the positive part of the quantum group $U_q^+(rak{so}_{2n+1})$, establishing a bijection between such subalgebras and integer sequences $(\theta_1, \dots, \theta_n)$ with $0 \leq \theta_k \leq 2n - 2k + 1$. The key result is that there are exactly $(2n)!!$ such subalgebras, matching the order of the Weyl group of type $B_n$, and the classification extends to the small Lusztig quantum group when $q^t = 1$, $t > 4$. The classification relies on PBW generators and a coproduct formula for skew-root vectors using group-like elements and structure constants.

ABSTRACT

We give a complete classification of right coideal subalgebras that contain all group-like elements for the quantum group $U_q^+(\frak{so}_{2n+1}),$ provided that $q$ is not a root of 1. If $q$ has a finite multiplicative order $t>4,$ this classification remains valid for homogeneous right coideal subalgebras of the small Lusztig quantum group $u_q^+(\frak{so}_{2n+1}).$ As a consequence, we determine that the total number of right coideal subalgebras that contain the coradical equals $(2n)!!,$ the order of the Weyl group defined by the root system of type $B_n.$

Motivation & Objective

• To classify all right coideal subalgebras of $U_q^+(\frak{so}_{2n+1})$ that contain the coradical, i.e., the span of group-like elements.
• To extend this classification to homogeneous right coideal subalgebras of the small Lusztig quantum group $u_q^+(\frak{so}_{2n+1})$ when $q$ has finite order $t > 4$.
• To establish a bijection between such subalgebras and integer sequences $\theta = (\theta_1, \dots, \theta_n)$ with $0 \leq \theta_k \leq 2n - 2k + 1$.
• To show that the total number of such subalgebras is $(2n)!!$, coinciding with the order of the Weyl group of type $B_n$.
• To generalize the classification to subalgebras whose intersection with the group-like elements forms a subgroup, not necessarily the full group.

Proposed method

• Construct a PBW basis for $U_q^+(\frak{so}_{2n+1})$ using skew-root vectors $u[k,m]$ defined via iterated quantum brackets.
• Derive an explicit coproduct formula for these PBW generators: $\Delta(u[k,m]) = u[k,m] \otimes 1 + g_{km} \otimes u[k,m] + \sum_{i=k}^{m-1} \tau_i(1 - q^{-2}) g_{ki} u[i+1,m] \otimes u[k,i]$, with $\tau_n = q$ and $\tau_i = 1$ otherwise.
• Define new PBW generators $\Phi^S(k,m)$ via recursive relations involving structure constants $\alpha_{km}^s$, ensuring compatibility with the coideal condition.
• Introduce the concept of $(k,m)$-regular sets $S$ to control the structure of $\Phi^S(k,m)$ and enable duality arguments.
• Define a root sequence $r({\bf U}) = (\theta_1, \dots, \theta_n)$ for a coideal subalgebra $\bf U$, where $\theta_k$ is the maximal $m$ such that $\Phi^S(k,m) \in \bf U$ and its degree is not a sum of other degrees in $\bf U$.
• Construct a subalgebra $\bf U_\theta$ from sequences $\theta$ using recursive definitions of sets $R_k$ and $T_k$, and show that $\bf U_\theta$ is a right coideal subalgebra containing the coradical.

Experimental results

Research questions

• RQ1 How many right coideal subalgebras of $U_q^+(\frak{so}_{2n+1})$ contain the coradical when $q$ is not a root of unity?
• RQ2 Can the classification of such subalgebras be extended to the small Lusztig quantum group $u_q^+(\frak{so}_{2n+1})$ when $q^t = 1$, $t > 4$?
• RQ3 Is there a canonical parametrization of all such coideal subalgebras via integer sequences $\theta_k$ with $0 \leq \theta_k \leq 2n - 2k + 1$?
• RQ4 Why does the number of such subalgebras equal $(2n)!!$, the order of the Weyl group of type $B_n$?
• RQ5 What is the structure of homogeneous right coideal subalgebras whose intersection with the group-like elements is a subgroup, not necessarily the full group?

Key findings

• There is a canonical bijection between the set of all right coideal subalgebras of $U_q^+(\frak{so}_{2n+1})$ containing the coradical and the set of integer sequences $\theta = (\theta_1, \dots, \theta_n)$ with $0 \leq \theta_k \leq 2n - 2k + 1$.
• The total number of such right coideal subalgebras is $(2n)!!$, which matches the order of the Weyl group of the root system of type $B_n$.
• The classification remains valid for homogeneous right coideal subalgebras of the small Lusztig quantum group $u_q^+(\frak{so}_{2n+1})$ when $q^t = 1$ and $t > 4$.
• Each such subalgebra $\bf U_\theta$ is generated by the group-like elements and elements $\Phi^{T_k}(k,m)$ for $m \in R_k$, where $R_k$ and $T_k$ are recursively defined sets.
• For any homogeneous right coideal subalgebra $\bf U$ of $U_q^+(\frak{so}_{2n+1})$ or $u_q^+(\frak{so}_{2n+1})$ with $\bf U \cap G$ a subgroup, $\bf U = k[\Omega] \cdot \bf U_\theta^1$ for some $\theta$ and subgroup $\Omega \subseteq G$.
• The subalgebra $\bf U_\theta^1$ is homogeneous, generated by $g_{km}^{-1} \Phi^S(k,m)$, and satisfies $\bf U_\theta^1 \cap G = \{1\}$, making it a minimal coideal subalgebra over the trivial group.