[Paper Review] The algebra $U^+_q$ and its alternating central extension $\mathcal U^+_q$

Paul Terwilliger|arXiv (Cornell University)|Jun 26, 2021

Algebraic structures and combinatorial models12 references4 citations

TL;DR

This paper establishes a generating function formula that expresses Damiani's PBW basis elements of the positive part $U^+_q$ of the quantum affine algebra $U_q(\widehat{\mathfrak{sl}}_2)$ in terms of the alternating generators of its central extension $\mathcal{U}^+_q$. The key contribution is a closed-form relation between the classical PBW basis and the new alternating generators via generating functions, resolving the structure of the extended algebra and enabling explicit computation of basis elements in the new framework.

ABSTRACT

Let $U^+_q$ denote the positive part of the quantized enveloping algebra $U_q(\widehat{\mathfrak{sl}}_2)$. The algebra $U^+_q$ has a presentation involving two generators $W_0$, $W_1$ and two relations, called the $q$-Serre relations. In 1993 I. Damiani obtained a PBW basis for $U^+_q$, consisting of some elements $\lbrace E_{n δ+ α_0} brace_{n=0}^\infty$, $\lbrace E_{n δ+ α_1} brace_{n=0}^\infty$, $\lbrace E_{n δ} brace_{n=1}^\infty$. In 2019 we introduced the alternating central extension $\mathcal U^+_q$ of $U^+_q$. We defined $\mathcal U^+_q$ by generators and relations. The generators, said to be alternating, are denoted $\lbrace \mathcal W_{-k} brace_{k=0}^\infty$, $\lbrace \mathcal W_{k+1} brace_{k=0}^\infty$, $ \lbrace \mathcal G_{k+1} brace_{k=0}^\infty$, $\lbrace \mathcal { ilde G}_{k+1} brace_{k=0}^\infty$. Let $\langle \mathcal W_0, \mathcal W_1 angle$ denote the subalgebra of $\mathcal U^+_q$ generated by $\mathcal W_0$, $\mathcal W_1$. It is known that there exists an algebra isomorphism $U^+_q o \langle \mathcal W_0, \mathcal W_1 angle$ that sends $W_0 \mapsto \mathcal W_0$ and $W_1 \mapsto \mathcal W_1$. Via this isomorphism we identify $U^+_q$ with $\langle \mathcal W_0, \mathcal W_1 angle$. In our main result, we express the Damiani PBW basis elements in terms of the alternating generators. We give the answer in terms of generating functions.

Motivation & Objective

• To express the Damiani PBW basis of $U^+_q$ in terms of the alternating generators of its central extension $\mathcal{U}^+_q$.
• To establish a generating function framework that links classical PBW basis elements to the new alternating generators in $\mathcal{U}^+_q$.
• To clarify the algebraic structure of $\mathcal{U}^+_q$ by embedding the classical $U^+_q$ via an isomorphism and extending its basis.

Proposed method

• The paper uses the isomorphism $\varphi: \mathcal{U}^+_q \to U^+_q \otimes \mathbb{F}[z_1,z_2,\ldots]$ to lift the classical PBW basis elements into the extended algebra.
• It defines generating functions $E^-(t) = \sum_{n=0}^\infty E_{n\delta + \alpha_0} t^n$, $E^+(t) = \sum_{n=0}^\infty E_{n\delta + \alpha_1} t^n$, and $E(t) = \sum_{n=0}^\infty E_{n\delta} t^n$ to encode the PBW basis elements.
• The main result is derived via generating function identities involving the alternating generators $\mathcal{W}_k$, $\mathcal{G}_k$, and $\widetilde{\mathcal{G}}_k$, particularly through the relations in equations (77)–(81).
• The derivation relies on known commutation relations among the alternating generators, as formalized in Lemma 14.2 and Corollary 12.5.
• Reduction rules for the PBW basis of $\mathcal{U}^+_q$ are used to reorder products of generators, ensuring consistency with the generating function expressions.
• The structure of the central extension $\mathcal{U}^+_q$ is exploited via the isomorphism $\varphi$, which maps the center to $1 \otimes \mathbb{F}[z_1,z_2,\ldots]$.

Experimental results

Research questions

• RQ1 How can the Damiani PBW basis of $U^+_q$ be expressed in terms of the alternating generators of its central extension $\mathcal{U}^+_q$?
• RQ2 What generating function relations govern the correspondence between classical PBW basis elements and the new alternating generators?
• RQ3 How do the commutation relations in $\mathcal{U}^+_q$ facilitate the reconstruction of the classical PBW basis?
• RQ4 What is the role of the central extension $\mathcal{U}^+_q$ in unifying the classical and extended algebraic structures?
• RQ5 Can the generating functions $E^-(t)$, $E^+(t)$, and $E(t)$ be rewritten using only the alternating generators and their relations?

Key findings

• The paper derives a generating function expression for the classical PBW basis elements of $U^+_q$ in terms of the alternating generators of $\mathcal{U}^+_q$, establishing a direct algebraic correspondence.
• The relation $\bigl{(}\widetilde{\mathcal{G}}(t)\bigr{)}^{-1}\mathcal{W}_0 = q^2\mathcal{W}_0\bigl{(}\widetilde{\mathcal{G}}(t)\bigr{)}^{-1} - q(q-q^{-1})\mathcal{W}^-(q^{-2}t)\bigl{(}\widetilde{\mathcal{G}}(q^{-2}t)\bigr{)}^{-1}\bigl{(}\widetilde{\mathcal{G}}(t)\bigr{)}^{-1}$ is proven as a key identity linking $\mathcal{W}_0$ and the $\widetilde{\mathcal{G}}$-generating function.
• Similarly, the identity $\bigl{(}\widetilde{\mathcal{G}}(t)\bigr{)}^{-1}\mathcal{W}_1 = q^{-2}\mathcal{W}_1\bigl{(}\widetilde{\mathcal{G}}(t)\bigr{)}^{-1} + q^{-1}(q-q^{-1})\mathcal{W}^+(q^2t)\bigl{(}\widetilde{\mathcal{G}}(q^2t)\bigr{)}^{-1}\bigl{(}\widetilde{\mathcal{G}}(t)\bigr{)}^{-1}$ is established as a fundamental commutation rule.
• The generating functions $E^-(t)$, $E^+(t)$, and $E(t)$ are shown to be expressible in terms of the alternating generators via the derived functional equations.
• The reduction rules in Lemma 14.3 allow systematic reordering of products of alternating generators, ensuring consistency with the PBW basis structure in $\mathcal{U}^+_q$.
• The isomorphism $\varphi: \mathcal{U}^+_q \to U^+_q \otimes \mathbb{F}[z_1,z_2,\ldots]$ enables the translation of classical basis elements into the extended algebra, with $\mathcal{W}_0$ and $\mathcal{W}_1$ mapping to $W_0 \otimes 1$ and $W_1 \otimes 1$ respectively.