The Formal Shift Operator on the Yangian Double

Abstract

Abstract Let ${\mathfrak{g}}$ be a symmetrizable Kac–Moody algebra with associated Yangian $Y_\hbar{\mathfrak{g}}$ and Yangian double $\textrm{D}Y_\hbar{\mathfrak{g}}$. 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 $\textrm{D}Y_\hbar{\mathfrak{g}}$ 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 $\textrm{D}Y_\hbar{\mathfrak{g}}$ 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 $\textrm{D}Y_\hbar{\mathfrak{g}}$ 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 $\textrm{D}Y_\hbar{\mathfrak{g}}$. Using these results, we obtain a Poincaré–Birkhoff–Witt theorem for $\textrm{D}Y_\hbar{\mathfrak{g}}$ applicable when ${\mathfrak{g}}$ is of finite type or of simply laced affine type.