The formal shift operator on the Yangian double

Abstract

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra with associated Yangian $Y_{\hslash }\mathfrak{g}$ and Yangian double $\mathrm{D}{Y}_{\hslash }\mathfrak{g}$. 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 $Y_{\hslash }\mathfrak{g}$ 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 $Y_{\hslash }\mathfrak{g}$ to the polynomial algebra $Y_{\hslash }\mathfrak{g}[z]$. We prove that ${\tau }_z$ uniquely extends to an algebra homomorphism ${\Phi }_z$ from the Yangian double $\mathrm{D}{Y}_{\hslash }\mathfrak{g}$ into the $\hslash$-adic closure of the algebra of Laurent series in $z^{-1}$ with coefficients in the Yangian $Y_{\hslash }\mathfrak{g}$. This induces, via evaluation at any point $c\in {\mathbb{C}}^{\times }$, a homomorphism from $\mathrm{D}{Y}_{\hslash }\mathfrak{g}$ 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 $\mathrm{D}{Y}_{\hslash }\mathfrak{g}$ and $Y_{\hslash }\mathfrak{g}$ and, as a corollary, we find that the Yangian $Y_{\hslash }\mathfrak{g}$ can be realized as a degeneration of the Yangian double $\mathrm{D}{Y}_{\hslash }\mathfrak{g}$. Using these results, we obtain a Poincaré-Birkhoff-Witt theorem for $\mathrm{D}{Y}_{\hslash }\mathfrak{g}$ applicable when $\mathfrak{g}$ is of finite type or of simply-laced affine type.

arXiv comment40 pages

References in corpus (13)

• The affine Yangian of ${\mathfrak{gl}}_1$ revisited
• Coproduct for the Yangian of an affine Kac-Moody algebra
• The cohomological Hall algebra of a preprojective algebra
• Yangians and quantum loop algebras
• Equivalences between three presentations of orthogonal and symplectic Yangians
• Shifted quantum affine algebras: integral forms in type $A$ (with appendices by Alexander Tsymbaliuk and Alex Weekes)
• Center of the quantum affine vertex algebra in type A
• Homomorphisms between different quantum toroidal and affine Yangian algebras
• Vertex Representations for Yangians of Kac-Moody algebras
• Double Yangian and the universal R-matrix
• Yangian doubles of classical types and their vertex representations
• The meromorphic R-matrix of the Yangian
• Classical limits of quantum toroidal and affine Yangian algebras

Cited by in corpus (6)

• Poles of finite-dimensional representations of Yangians
• The meromorphic R-matrix of the Yangian
• Affine $ı$quantum groups and twisted Yangians in Drinfeld presentations
• Double Yangian and reflection algebras of the Lie superalgebra ${\mathfrak{gl}}_{m|n}$
• On a conjecture of Khoroshkin and Tolstoy
• The restricted quantum double of the Yangian