Compatibility between truncation and coproducts for quantum affine algebra and Yangian of $\mathfrak{sl}_2(\mathbb{C})$

TL;DR

The paper proves that the Drinfeld–Jimbo coproducts for $\mathfrak{sl}_2$ Yangians and quantum affine algebras factor through their truncated quotients, i.e., there are algebra morphisms $Y^{ab}\to Y^a\otimes Y^b$ and $U_q^{ab}\to U_q^a\otimes U_q^b$. Central to the construction is the GKLO $A$-series, with explicit coproduct formulas derived for $A(z)$ (and $A^-(z)$ in the quantum case), together with a multiplicative reformulation $A^a(z)=a^*(z)A(z)$ that preserves truncation. The method combines auxiliary $S(z)$-series and $q$-exponentials to obtain exact coproduct expressions, which are then shown to land in the corresponding truncated algebras, enabling factorization of tensor products through truncations. The results extend to shifted versions of Yangians and quantum affine algebras and have implications for representations in KTWWY and BFN contexts, by ensuring that tensor products factor through truncated categories. Overall, the work provides a concrete, computable bridge between infinite-dimensional quantum groups and their finite truncated quotients in the $\mathfrak{sl}_2$ setting, with potential for generalization to other types.

Abstract

We prove that the standard Drinfeld-Jimbo coproducts for Yangians and quantum affine algebras factorize through their truncated quotients in the case of $\mathfrak{sl}_2(\mathbb{C})$. As an auxiliary result, we give formulas for the coproduct of the truncation series in both cases.

Theorems & Definitions (39)

• Definition 1
• remark 1
• Definition 2
• remark 2
• Definition 3
• Proposition 4
• remark 3
• Proposition 5
• Proposition 6
• proof