论文标题
来自抗主导地移动扬吉亚人和量子仿射代数的松弛矩阵：a型
Lax matrices from antidominantly shifted Yangians and quantum affine algebras: A-type
论文作者
论文摘要
我们构建了一个$ gl_n $ Rational和Trigonometric Lax矩阵的家庭，由$λ^+$ - 估价的divisors $ d $ on $ \ mathbb {p}^1 $。为此，我们研究了转移的Drinfeld Yangians $y_μ（\ Mathfrak {gl} _n）$和量子仿射代数$ u_ {μ^+，μ^ - }（l \ mathfrak {gl} _n）$我们的主要观察结果是，当$μ$（分别分别为$μ^+$和$μ^ - $）是抗利用型牛皮时，这两个代数都允许RTT型实现。我们证明，$ t_d（z）$在$ z $（最多是理性因素）中是多项式，并为$ z $中的线性获得明确的简单公式。这概括了三角和较高$ z $ - $度的方向的前两位线性有理洛杉矶力矩阵的最新构建。此外，我们表明所有$ t_d（z）$都是由$ \ {\ fiftty \} $所支持的$ d $参数的标准化限制（在理性情况下）或$ \ {0，\ fist \ infty \} $（在三角案例中）。 RTT方法为在$ \ mathfrak {sl} _n $的转移Yangians和量子仿射代数上构建同构同构提供了概念和基本证据。最后，我们建立了一定的显式线性松弛矩阵和众所周知的抛物线抛物线Gelfand-tsetlin公式之间的密切关系。
We construct a family of $GL_n$ rational and trigonometric Lax matrices $T_D(z)$ parametrized by $Λ^+$-valued divisors $D$ on $\mathbb{P}^1$. To this end, we study the shifted Drinfeld Yangians $Y_μ(\mathfrak{gl}_n)$ and quantum affine algebras $U_{μ^+,μ^-}(L\mathfrak{gl}_n)$, which slightly generalize their $\mathfrak{sl}_n$-counterparts. Our key observation is that both algebras admit the RTT type realization when $μ$ (respectively, $μ^+$ and $μ^-$) 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.
