[Paper Review] On the Quasi-Hopf structure of deformed double Yangians

D. Arnaudon, Jean Avan+3 more|arXiv (Cornell University)|Jan 6, 2000

Algebraic structures and combinatorial models14 references3 citations

TL;DR

This paper establishes the quasi-Hopf algebra structure of deformed double Yangians $\mathcal{D}Y_r^{V6}$, $\mathcal{D}Y_r^{V8}$, and $\mathcal{D}Y_r^{F}$ by constructing universal twists from the centrally extended double Yangian $\mathcal{D}Y(sl(2))_c$. Using a systematic method based on linear difference equations and evaluation representations, it proves that these three algebras are equivalent as quasi-Hopf algebras under different gauge choices, with their $R$-matrices related via universal twist transformations satisfying the shifted cocycle condition.

ABSTRACT

We construct universal twists connecting the centrally extended double Yangian DY(sl(2))_c with deformed double Yangians DY_r(sl(2))_c, thereby establishing the quasi-Hopf structures of the latter.

Motivation & Objective

• To establish the quasi-Hopf algebra structure of deformed double Yangians $\mathcal{D}Y_r^{V6}$, $\mathcal{D}Y_r^{V8}$, and $\mathcal{D}Y_r^{F}$ by constructing universal twists from the central extension of the double Yangian $\mathcal{D}Y(sl(2))_c$.
• To demonstrate that these three deformed algebras are isomorphic as quasi-Hopf algebras via universal twist transformations, despite differing in generator presentations and $R$-matrix realizations.
• To provide a consistent universal framework for the $R$-matrix transformations between these algebras by deriving and solving a linear difference equation for the twist operator.

Proposed method

• Construct a universal twist $\mathcal{F}$ connecting $\mathcal{D}Y$ to $\mathcal{D}Y_r^{V6}$ by identifying a twist-like action in representation space and promoting it to a universal level via an infinite product representation.
• Derive a linear difference equation for the twist operator based on its representation, which is then promoted to a universal linear equation governing the universal twist $\mathcal{F}$.
• Prove that the solution of the universal linear equation satisfies the shifted cocycle condition, thereby ensuring that the transformed $R$-matrix $\mathcal{R}^\mathcal{F}$ satisfies the shifted Yang–Baxter equation.
• Identify the $R$-matrix connection between $\mathcal{D}Y_r^{V6}$ and $\mathcal{D}Y_r^{V8}$ as a universal coboundary twist, with the twist element $g = \exp\left(\frac{\pi}{2}(f_0 - e_0)\right)$, whose evaluation realizes the $K$-matrix transformation.
• Establish the $R$-matrix connection between $\mathcal{D}Y_r^{V6}$ and $\mathcal{D}Y_r^{F}$ via a second universal coboundary twist using $g' = \exp\left(\frac{h_1}{2r}\right)$, leading to a $K^{(6)}$-matrix transformation.
• Verify that all twisted $R$-matrices satisfy the shifted Yang–Baxter equation and that the resulting algebras $\mathcal{D}Y_r^{V6}$, $\mathcal{D}Y_r^{V8}$, and $\mathcal{D}Y_r^{F}$ are isomorphic quasi-Hopf algebras under different gauge choices.

Experimental results

Research questions

• RQ1 How can the quasi-Hopf structure of the deformed double Yangians $\mathcal{D}Y_r^{V6}$, $\mathcal{D}Y_r^{V8}$, and $\mathcal{D}Y_r^{F}$ be systematically constructed from the universal $R$-matrix of the central extension $\mathcal{D}Y(sl(2))_c$?
• RQ2 What universal twist operators connect $\mathcal{D}Y$ to $\mathcal{D}Y_r^{V6}$, $\mathcal{D}Y_r^{V8}$, and $\mathcal{D}Y_r^{F}$, and how do they satisfy the shifted cocycle condition?
• RQ3 Are the three deformed algebras $\mathcal{D}Y_r^{V6}$, $\mathcal{D}Y_r^{V8}$, and $\mathcal{D}Y_r^{F}$ isomorphic as quasi-Hopf algebras, and if so, what is the universal twist that realizes this isomorphism?
• RQ4 Can the $R$-matrix transformations between $\mathcal{D}Y_r^{V6}$ and $\mathcal{D}Y_r^{V8}$, and between $\mathcal{D}Y_r^{V6}$ and $\mathcal{D}Y_r^{F}$, be derived from universal coboundary twists with explicit algebraic elements?
• RQ5 What is the role of the linear difference equation in constructing the universal twist and ensuring consistency with the universal $R$-matrix transformation rule?

Key findings

• The universal twist $\mathcal{F}$ connecting $\mathcal{D}Y$ to $\mathcal{D}Y_r^{V6}$ is constructed as an infinite product solution to a linear difference equation, and it satisfies the shifted cocycle condition, confirming the quasi-Hopf structure of $\mathcal{D}Y_r^{V6}$.
• The $R$-matrix of $\mathcal{D}Y_r^{V8}$ is obtained from that of $\mathcal{D}Y_r^{V6}$ via a universal coboundary twist using $g = \exp\left(\frac{\pi}{2}(f_0 - e_0)\right)$, proving that $\mathcal{D}Y_r^{V8}$ is isomorphic to $\mathcal{D}Y_r^{V6}$ as a quasi-Hopf algebra.
• The $R$-matrix of $\mathcal{D}Y_r^{F}$ is related to that of $\mathcal{D}Y_r^{V6}$ by a second universal coboundary twist with $g' = \exp\left(\frac{h_1}{2r}\right)$, showing that $\mathcal{D}Y_r^{F}$ is also isomorphic to $\mathcal{D}Y_r^{V6}$ as a quasi-Hopf algebra.
• All three deformed algebras $\mathcal{D}Y_r^{V6}$, $\mathcal{D}Y_r^{V8}$, and $\mathcal{D}Y_r^{F}$ are shown to be different gauge realizations of the same underlying quasi-Hopf algebra, with the twist operators providing explicit isomorphisms.
• The universal $R$-matrix of each deformed algebra satisfies the shifted Yang–Baxter equation, confirming their consistent quasi-Hopf algebra structure, with the universal twist $\mathcal{F}$ inducing the correct coproduct deformation $\Delta^\mathcal{F}(x) = \mathcal{F} \Delta(x) \mathcal{F}^{-1}$.