论文标题
Yangian Double上的正式班次操作员
The formal shift operator on the Yangian double
论文作者
论文摘要
令$ \ mathfrak {g} $为一个对称的kac-moody代数，与关联的yangian $ y_ \ hbar \ hbar \ mathfrak {g} $和yangian double $ \ mathrm {d} y___ \ hbar \ hbar \ hbar \ mathfrak {g} $。扬吉人理论的基本结果是，对于\ mathbb {c} $中的每个$ c \，都有$ y_ \ y_ \ y_ \ hbar \ hbar \ hbar \ hbar \ hbar \ hbar \ mathfrak {g} $的自动形态$τ_c$，与transporation $ t \ t \ mapsto t \ mapsto t+c $相对应。用正式参数$ z $替换$ c $，从$ y_ \ hbar \ hbar \ mathfrak {g} $到polyenmial algebra $ y_ \ y_ \ hbar \ hbar \ hbar \ mathfrak \ mathfrak {g} [z] $。 我们证明$τ_z$唯一扩展到来自扬式的double $ \ mathrm {d} y_ \ hbar \ hbar \ mathfrak {g} $的代数同构$φ_z$ $ y_ \ hbar \ Mathfrak {g} $。这可以通过评估在\ Mathbb {c}^\ times $中通过评估，这是$ \ mathrm {d} y_ \ hbar \ hbar \ mathfrak {g} $的同型同构，进入Yangian的分级。 We show that each such homomorphism gives rise to an isomorphism between completions of $\mathrm{D}Y_\hbar\mathfrak{g}$ and $Y_\hbar\mathfrak{g}$ and, as a corollary, we find that the Yangian $Y_\hbar\mathfrak{g}$ can be realized as a degeneration of the yangian double $ \ mathrm {d} y_ \ hbar \ hbar \ mathfrak {g} $。使用这些结果，我们获得了$ \ mathrm {d} y_ \ hbar \ hbar \ mathfrak {g} $的Poincaré-birkhoff-witt定理，当$ \ mathfrak {g} $是有限类型或简单带lace lace facine类型时。
Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra with associated Yangian $Y_\hbar\mathfrak{g}$ and Yangian double $\mathrm{D}Y_\hbar\mathfrak{g}$. An elementary result of fundamental importance to the theory of Yangians is that, for each $c\in \mathbb{C}$, there is an automorphism $τ_c$ of $Y_\hbar\mathfrak{g}$ corresponding to the translation $t\mapsto t+c$ of the complex plane. Replacing $c$ by a formal parameter $z$ yields the so-called formal shift homomorphism $τ_z$ from $Y_\hbar\mathfrak{g}$ to the polynomial algebra $Y_\hbar\mathfrak{g}[z]$. We prove that $τ_z$ uniquely extends to an algebra homomorphism $Φ_z$ from the Yangian double $\mathrm{D}Y_\hbar\mathfrak{g}$ into the $\hbar$-adic closure of the algebra of Laurent series in $z^{-1}$ with coefficients in the Yangian $Y_\hbar\mathfrak{g}$. This induces, via evaluation at any point $c\in \mathbb{C}^\times$, a homomorphism from $\mathrm{D}Y_\hbar\mathfrak{g}$ into the completion of the Yangian with respect to its grading. We show that each such homomorphism gives rise to an isomorphism between completions of $\mathrm{D}Y_\hbar\mathfrak{g}$ and $Y_\hbar\mathfrak{g}$ and, as a corollary, we find that the Yangian $Y_\hbar\mathfrak{g}$ can be realized as a degeneration of the Yangian double $\mathrm{D}Y_\hbar\mathfrak{g}$. Using these results, we obtain a Poincaré-Birkhoff-Witt theorem for $\mathrm{D}Y_\hbar\mathfrak{g}$ applicable when $\mathfrak{g}$ is of finite type or of simply-laced affine type.
