The formal shift operator on the Yangian double

TL;DR

This paper extends the formal shift automorphism from Yangians to their doubles, establishing isomorphisms between their completions and showing Yangians as degenerations of Yangian doubles, with applications to PBW theorems.

Contribution

It introduces a unique extension of the formal shift homomorphism to Yangian doubles and demonstrates their structural relationship and degenerations.

Findings

Extension of $ au_z$ to $ ext{D}Y_ ext{hbar}rak{g}$ as $ ext{PBW}$-type isomorphisms.

Realization of Yangian $Y_ ext{hbar}rak{g}$ as a degeneration of the Yangian double.

Applicable PBW theorem for $ ext{D}Y_ ext{hbar}rak{g}$ in finite and affine types.

Abstract

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 $\tau_c$ 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 $\tau_z$ from $Y_\hbar\mathfrak{g}$ to the polynomial algebra $Y_\hbar\mathfrak{g}[z]$. We prove that $\tau_z$ uniquely extends to an algebra homomorphism $\Phi_z$ 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 Poincar\'{e}-Birkhoff-Witt theorem for $\mathrm{D}Y_\hbar\mathfrak{g}$ applicable when $\mathfrak{g}$ is of finite type or of simply-laced affine type.