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Mathematics, Representation Theory, Mathematical Physics
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.
