[Paper Review] Lax matrices from antidominantly shifted Yangians and quantum affine algebras: A-type

Rouven Frassek, Vasily Pestun+1 more|arXiv (Cornell University)|Jan 14, 2020

Algebraic structures and combinatorial models15 references8 citations

TL;DR

This paper constructs GLn rational and trigonometric Lax matrices TD(z) parametrized by Λ+-valued divisors D on P¹ using antidominantly shifted Drinfeld Yangians and quantum affine algebras of type A. It establishes an RTT realization for these algebras when the shift parameters are antidominant, proving that TD(z) are polynomial in z (up to rational factors) and providing explicit formulas for linear cases. The key contribution is a conceptual, elementary proof of coproduct homomorphisms on shifted algebras via the RTT approach, generalizing previous results and linking to Gelfand-Tsetlin bases.

ABSTRACT

We construct a family of $GL_n$ rational and trigonometric Lax matrices $T_D(z)$ parametrized by $\Lambda^+$-valued divisors $D$ on $\mathbb{P}^1$. To this end, we study the shifted Drinfeld Yangians $Y_\mu(\mathfrak{gl}_n)$ and quantum affine algebras $U_{\mu^+,\mu^-}(L\mathfrak{gl}_n)$, which slightly generalize their $\mathfrak{sl}_n$-counterparts. Our key observation is that both algebras admit the RTT type realization when $\mu$ (respectively, $\mu^+$ and $\mu^-$) are antidominant coweights. We prove that $T_D(z)$ are polynomial in $z$ (up to a rational factor) and obtain explicit simple formulas for those linear in $z$. This generalizes the recent construction by the first two authors of linear rational Lax matrices in both trigonometric and higher $z$-degree directions. Furthermore, we show that all $T_D(z)$ are normalized limits of those parametrized by $D$ supported away from $\{\infty\}$ (in the rational case) or $\{0,\infty\}$ (in the trigonometric case). The RTT approach provides conceptual and elementary proofs for the construction of the coproduct homomorphisms on shifted Yangians and quantum affine algebras of $\mathfrak{sl}_n$, previously established via rather tedious computations. Finally, we establish a close relation between a certain collection of explicit linear Lax matrices and the well-known parabolic Gelfand-Tsetlin formulas.

Motivation & Objective

• To construct rational and trigonometric Lax matrices TD(z) for GLn parametrized by Λ+-valued divisors D on P¹.
• To establish an RTT-type realization of antidominantly shifted Drinfeld Yangians Yµ(gln) and quantum affine algebras Uµ+,µ−(Lgln), generalizing previous results.
• To prove that the constructed Lax matrices TD(z) are polynomial in z (up to rational factors), with explicit formulas for linear cases.
• To provide conceptual and elementary proofs for coproduct homomorphisms on shifted Yangians and quantum affine algebras, previously established via tedious computations.
• To establish a direct connection between explicit linear Lax matrices and the classical parabolic Gelfand-Tsetlin formulas.

Proposed method

• The authors use shifted Drinfeld Yangians Yµ(gln) and quantum affine algebras Uµ+,µ−(Lgln) with antidominant coweight shifts µ, µ+ and µ−.
• They introduce an RTT realization for these algebras, defining generators T±(z) satisfying RTT relations, which allows for a systematic construction of Lax matrices.
• The Lax matrices TD(z) are constructed via homomorphisms ΨD from the RTT algebra to matrix algebras, ensuring prescribed singularities at divisor points D.
• The normalized limit construction is used to show that all TD(z) arise as limits of matrices supported away from ∞ (rational) or {0, ∞} (trigonometric), providing a physical interpretation.
• The coproduct homomorphisms are derived directly from the RTT structure, with ∆rtt defined by T±(z) ↦ T±(z) ⊗ T±(z), and extended to shifted algebras.
• Explicit formulas for quantum determinants qdet TD(z) are computed for small n, verifying consistency with known cases and providing concrete examples.

Experimental results

Research questions

• RQ1 How can Lax matrices TD(z) be systematically constructed for GLn with prescribed singularities encoded by a Λ+-valued divisor D on P¹?
• RQ2 Under what conditions on the shift parameters (antidominant coweights) does the RTT realization hold for shifted Drinfeld Yangians and quantum affine algebras of type A?
• RQ3 What is the relationship between the constructed Lax matrices and the classical Gelfand-Tsetlin bases of parabolic Verma modules?
• RQ4 How do the coproduct homomorphisms on shifted Yangians and quantum affine algebras arise naturally from the RTT structure?
• RQ5 Can the normalized limit construction recover all Lax matrices from those supported away from ∞ or {0, ∞}, and what is its physical significance in gauge theory?

Key findings

• The Lax matrices TD(z) are polynomial in z up to a rational factor, and explicit formulas are derived for those linear in z, generalizing earlier constructions.
• For n=2, six explicit linear trigonometric Lax matrices are computed with quantum determinants qdet TD(z) = v²z², z, v⁻², v²z(z−v⁻²x₁), z−v⁻²x₁, and v²(z−v⁻²x₁)(z−v⁻²x₂), respectively.
• The RTT realization provides a conceptual and elementary proof of coproduct homomorphisms on shifted Yangians and quantum affine algebras, replacing earlier lengthy computations.
• All Lax matrices TD(z) arise as normalized limits of matrices supported away from ∞ (rational) or {0, ∞} (trigonometric), consistent with physical expectations from N=2 quiver gauge theories.
• The construction establishes a direct link between explicit linear Lax matrices and the parabolic Gelfand-Tsetlin formulas, confirming a long-standing expectation.
• The coproduct homomorphisms on the shifted algebras are explicitly computed and shown to coincide with known results, providing a simpler and more conceptual derivation.