[Paper Review] Elliptic Quantum Groups U_{q,p}(gl_N) and E_{q,p}(gl_N)

Hitoshi Konno|arXiv (Cornell University)|Mar 14, 2016

Algebraic structures and combinatorial models3 citations

TL;DR

This paper establishes an isomorphism between the Drinfeld realization $U_{q,p}( ext{gl}_N)$ and a central extension $E_{q,p}( ext{gl}_N)$ of the FRST formulation of elliptic quantum groups, using a topological algebra structure over formal power series in $p$. The key result is a precise correspondence between Drinfeld generators and $L$-operators via Gauss decomposition and $q$-determinant formulas, resolving long-standing isomorphism questions in elliptic quantum group theory.

ABSTRACT

We reformurate a central extension of Felder's elliptic quantum group in the FRST formulation as a topological algebra E_{q,p}(gl_N) over the ring of formal power series in p. We then discuss the isomorphism between E_{q,p}(gl_N) and the elliptic algebra U_{q,p}(gl_N) of the Drinfeld realization. An evaluation H-algebra homomorphism from U_{q,p}(gl_N) to a dynamical extension of the quantum affine algebra U_q(gl_N) resolves the problem into the one discussed by Ding and Frenkel in the trigonometric case. We also provide some useful formulas for the elliptic quantum determinants.

Motivation & Objective

• To resolve the long-standing open problem of establishing an isomorphism between the Drinfeld realization $U_{q,p}( ext{gl}_N)$ and the FRST formulation of elliptic quantum groups.
• To reformulate the FRST-type elliptic quantum group $E_{ au, u}( ext{gl}_N)$ as a topological algebra $E_{q,p}( ext{gl}_N)$ over formal power series in $p$.
• To construct a central extension of the FRST formulation compatible with the Hopf algebroid structure, enabling infinite-dimensional representation theory.
• To provide explicit formulas for elliptic quantum determinants and relate them to Drinfeld generators via Gauss decomposition.

Proposed method

• Reformulate the FRST-type elliptic quantum group $E_{ au, u}( ext{gl}_N)$ as a topological algebra $E_{q,p}( ext{gl}_N)$ over $\mathbb{C}[[p]]$ using $p$-adic topology.
• Define the central extension of $E_{ au, u}( ext{gl}_N)$ using the framework of [37, 36], resulting in $E_{q,p}( ext{gl}_N)$ with well-defined relations in the $p$-adic topology.
• Establish the isomorphism between $U_{q,p}( ext{gl}_N)$ and $E_{q,p}( ext{gl}_N)$ via an evaluation $H$-algebra homomorphism to a dynamical extension of $U_q(\widehat{\text{gl}}_N)$.
• Use Gauss decomposition of the $L$-operator $\widehat{L}^+(u)$ into $\mathcal{F}_{a,b}(u)\mathcal{K}_{a,b}(u)\mathcal{E}_{a,b}(u)$ to express Drinfeld generators in terms of matrix elements and $q$-determinants.
• Derive explicit formulas for $K^+_a(u)$, $E^+_{a,b}(u)$, and $F^+_{a,b}(u)$ in terms of $q$-determinants of submatrices of $\widehat{L}^+(u)$.
• Prove that the $q$-determinant ${q}\text{det}\widehat{L}^+(u)$ lies in the center of $E_{q,p}$(\text{gl}_N)$ via commutativity with all generators.

Experimental results

Research questions

• RQ1 Is there a canonical isomorphism between the Drinfeld realization $U_{q,p}(\widehat{\text{gl}}_N)$ and the FRST-type elliptic quantum group $E_{ au, u}(\text{gl}_N)$?
• RQ2 Can the FRST formulation of elliptic quantum groups be promoted to a central extension that supports infinite-dimensional representations?
• RQ3 How can Drinfeld generators be explicitly reconstructed from the $L$-operators in the FRST formulation using $q$-determinants?
• RQ4 What is the role of the $q$-determinant in the center of the extended FRST algebra $E_{q,p}(\text{gl}_N)$?
• RQ5 How does the evaluation homomorphism from $U_{q,p}(\text{gl}_N)$ to a dynamical quantum affine algebra resolve the isomorphism problem?

Key findings

• The algebra $E_{q,p}(\text{gl}_N)$ is defined as a topological algebra over $\mathbb{C}[[p]]$, ensuring well-defined relations in the $p$-adic topology.
• An explicit isomorphism is established between $U_{q,p}(\text{gl}_N)$ and $E_{q,p}(\text{gl}_N)$ via an evaluation $H$-algebra homomorphism to a dynamical extension of $U_q(\widehat{\text{gl}}_N)$.
• The Drinfeld generators $K^+_a(u)$, $E^+_{a,b}(u)$, and $F^+_{a,b}(u)$ are reconstructed from the $L$-operator via Gauss decomposition and $q$-determinant formulas.
• The $q$-determinant ${q}\text{det}\widehat{L}^+(u)$ is shown to be central in $E_{q,p}(\text{gl}_N)$, with $q$-det$\widehat{L}^+(u) = \mathcal{N}_N K(u)$, where $K(u) = \prod_{l=1}^N K^+_l(u - l + 1)$.
• The formulas for $K^+_a(u)$, $E^+_{a,b}(u)$, and $F^+_{a,b}(u)$ are derived as $K^+_a(u) = \frac{{q}\text{det}\widehat{L}^+(u)_{a,a}}{{q}\text{det}\widehat{L}^+(u-1)_{a+1,a+1}} \cdot \frac{1}{{\cal N}'_{N-a-1}}$, and similar expressions for $E^+_{a,b}(u)$ and $F^+_{a,b}(u)$ in terms of $q$-determinants of submatrices.
• The center of $E_{q,p}(\text{gl}_N)$ contains the $q$-determinant, and $K(u)$ commutes with all generators due to the commutation relations of the elliptic currents.