论文标题
Yangians和Kirillov-Reshetikhin Crystals的Bethe子代理
Bethe subalgebras in Yangians and Kirillov-Reshetikhin crystals
论文作者
论文摘要
令$ \ mathfrak {g} $为一个复杂的简单有限尺寸lie代数，$ g $为lie代数$ \ mathfrak {g} $的伴随谎言组。对于g $中的每一个$ c \，一个人都可以在yangian $ y（\ mathfrak {g}）$中关联一个交换性sibalgebra $ b（c）$，该$负责（广义）$ xxx $ heisenberg磁铁链的积分。使用Arxiv的方法：1708.05105，我们在Kirillov-Reshetikhin $ y（\ Mathfrak {g}）$ a $ $ a $ a $中构建了$ b（c）$ $ b（c）$ $ b（c）$的天然结构。我们猜想，这种构造存在于任意$ \ mathfrak {g} $，并给予Kirillov-Reshetikhin Crystals。我们的主要技术工具是扬吉亚人的伯特伯氏骨的变性，以均为$ \ mathcal {a}_χ^{\ mathrm {u}} $的普遍包裹在当前lie代数的普遍包膜中$ \ mathfrak {g} $（并且具有独立的兴趣）。我们表明，这些子代数来自Feigin，Frenkel和Toledano Laredo在Arxiv中的临界水平上的Feigin-Frenkel中心：Math/0612798。这可以证明我们在$ a $ a $中的仿射晶体确实是基里洛夫·雷希提金（Kirillov-reshetikhin），它通过还原为已知的不均匀gaudin模型的光谱上的晶体结构（arxiv：1708.05105）。
Let $\mathfrak{g}$ be a complex simple finite dimensional Lie algebra and $G$ be the adjoint Lie group with the Lie algebra $\mathfrak{g}$. To every $C \in G$ one can associate a commutative subalgebra $B(C)$ in the Yangian $Y(\mathfrak{g})$, which is responsible for the integrals of the (generalized) $XXX$ Heisenberg magnet chain. Using the approach of arXiv:1708.05105, we construct a natural structure of affine crystals on spectra of $B(C)$ in Kirillov-Reshetikhin $Y(\mathfrak{g})$-modules in type $A$. We conjecture that such a construction exists for arbitrary $\mathfrak{g}$ and gives Kirillov-Reshetikhin crystals. Our main technical tool is the degeneration of Bethe subalgebras in the Yangian to commutative subalgebras $\mathcal{A}_χ^{\mathrm{u}}$ in the universal enveloping of the current Lie algebra, $U(\mathfrak{g}[t])$, which depend on the parameter $χ$ from the Lie algebra $\mathfrak{g}$ (and are of independent interest). We show that these subalgebras come from the Feigin-Frenkel center on the critical level as described by Feigin, Frenkel and Toledano Laredo in arXiv:math/0612798. This allows to prove that our affine crystals in type $A$ are indeed Kirillov-Reshetikhin by reducing to the crystal structure on the spectra of inhomogeneous Gaudin model which is already known (arXiv:1708.05105).
