[Paper Review] Right coideal subalgebras in $U_q(\frak{sl}_{n+1}).$

V. K. Kharchenko, A.V. Lara Sagahon|ArXiv.org|Oct 11, 2007

Algebraic structures and combinatorial models40 references3 citations

TL;DR

This paper provides a complete classification of right coideal subalgebras containing the coradical in the multiparameter quantum group $U_q(\mathfrak{sl}_{n+1})$ when $q$ is not a root of unity, using a novel construction via PBW-bases and character Hopf algebra techniques. The key result is that for each subgroup $\Sigma$ of the group of group-like elements, there are exactly $(n+1)!$ homogeneous right coideal subalgebras with $U \cap G = \Sigma$, generalizing to finite-order $q > 2$ in the Lusztig quantum group $u_q(\mathfrak{sl}_{n+1})$. The classification is parametrized by integer sequences $\theta = (\theta_1,\dots,\theta_n)$ with $0 \leq \theta_i \leq n-i+1$. The paper also establishes conditions under which such subalgebras generate quantum symmetric pairs via commutator closure.

ABSTRACT

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 $Σ$ 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 $U\cap G=Σ.$ 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].

Motivation & Objective

• To classify all right coideal subalgebras of the multiparameter quantum group $U_q(\mathfrak{sl}_{n+1})$ that contain the coradical (i.e., all group-like elements), under the condition that $q$ is not a root of unity.
• To extend this classification to the Lusztig quantum group $u_q(\mathfrak{sl}_{n+1})$ when $q$ has finite multiplicative order $t > 2$, focusing on homogeneous right coideal subalgebras.
• To provide a unified framework using character Hopf algebras and noncommutative differential calculus to describe these subalgebras via PBW-bases over the coradical.
• To establish a one-to-one correspondence between such subalgebras and integer sequences $\theta = (\theta_1, \dots, \theta_n)$ with $0 \leq \theta_i \leq n-i+1$, enabling explicit construction of generators.

Proposed method

• The authors use the bosonisation construction $H = A \#{\bf k}[G]$ where $A$ is a quantum symmetric algebra and $G$ is the group of group-like elements, allowing the use of noncommutative differential calculus.
• They define PBW-bases over the coradical for right coideal subalgebras using iterated brackets $[u,v] = uv - \chi^u(g_v)vu$ and nested commutators $u[i,j] = [[\dots[x_i,x_{i+1}],\dots],x_j]$, forming the generators $\Psi^{T_k}(k,m)$.
• A sequence $\theta = (\theta_1,\dots,\theta_n)$ parametrizes each subalgebra via recursive definitions of sets $R_k$ and $T_k$, where $R_k$ contains indices satisfying specific inclusion and exclusion conditions relative to $T_{s+1}$ for $s \in R_k \setminus \{n\}$.
• The construction ensures that each generator $\Psi^{T_k}(k,m)$ lies in the right coideal subalgebra and spans it over the coradical $\mathbf{k}[G]$, with the PBW-basis providing a canonical form.
• The paper proves that any right coideal subalgebra $U \supseteq \mathbf{k}[G]$ arises as $U_\theta$ for some $\theta$, using the structure of $U^+_\theta$ and $U^-_{\theta'}$ and their commutator closure.
• The key technical tool is the ad-invariance and degree-reduction argument via ad-identities (2.9), (2.10), and the use of subdiagram decompositions to verify that $[U^+_\theta, U^-_{\theta'}] \subseteq U$ if and only if two conditions on $\theta$ and $\theta'$ are satisfied.

Experimental results

Research questions

• RQ1 How can all right coideal subalgebras of $U_q(\mathfrak{sl}_{n+1})$ containing the coradical be classified when $q$ is not a root of unity?
• RQ2 What is the precise parametrization of such subalgebras in terms of combinatorial data, and how does it relate to the group of group-like elements?
• RQ3 Under what conditions do two such subalgebras $U^+_\theta$ and $U^-_{\theta'}$ generate a subalgebra closed under the commutator bracket?
• RQ4 How does the classification extend to the finite-dimensional Lusztig quantum group $u_q(\mathfrak{sl}_{n+1})$ when $q$ has finite order $t > 2$?
• RQ5 What is the role of the PBW-basis over the coradical in characterizing invariants and structural properties of these coideal subalgebras?

Key findings

• For each subgroup $\Sigma$ of the group $G$ of group-like elements in $U_q(\mathfrak{sl}_{n+1})$, there are exactly $(n+1)!$ distinct homogeneous right coideal subalgebras $U$ such that $U \cap G = \Sigma$, provided $q$ is not a root of unity.
• The classification of right coideal subalgebras containing the coradical is in one-to-one correspondence with integer sequences $\theta = (\theta_1, \dots, \theta_n)$ satisfying $0 \leq \theta_i \leq n-i+1$, with each subalgebra $U_\theta$ generated by elements $\Psi^{T_k}(k,m)$ for $m \in R_k$.
• The generators $\Psi^{T_k}(k,m)$ are defined via iterated commutators $[\dots[x_k, x_{k+1}], \dots, x_m]$, where the bracket structure depends on the character $\chi^u(g_v)$, and they form a PBW-basis over $\mathbf{k}[G]$.
• A right coideal subalgebra $U$ is closed under the commutator bracket with $U^+_\theta$ and $U^-_{\theta'}$ if and only if the sequences $\theta$ and $\theta'$ satisfy two specific conditions: $\tilde{\theta}_k \notin T_{k'}$ for $k' \in R_k \setminus \{n\}$, and $T_k \cap T'_{k'} = \emptyset$ for certain indices.
• When $q$ has finite multiplicative order $t > 2$, the classification remains valid for homogeneous right coideal subalgebras of the Lusztig quantum group $u_q(\mathfrak{sl}_{n+1})$, extending the result to finite-dimensional settings.
• The paper establishes that the set of all right coideal subalgebras containing the coradical is finite, due to the finiteness of the parametrizing sequences $\theta$ and the PBW-basis structure.