Inflations for representations of shifted quantum affine algebras

Abstract

Fix a finite-dimensional simple Lie algebra $g$ and let $g_J\subseteq g$ be a Lie subalgebra coming from a Dynkin diagram inclusion. Then, the corresponding restriction functor is not essentially surjective on finite-dimensional simple $g_J$–modules.

In this article, we study Finkelberg–Tsymbaliuk's shifted quantum affine algebras $U_q^{\mu }\left(g\right)$ and the associated categories $O^{\mu }$ (defined by Hernandez). In particular, we introduce natural subalgebras $U_q^{\nu }\left(g_J\right)\subseteq U_q^{\mu }\left(g\right)$ and obtain a functor $R_J$ from $O^{sh}={⨁}_{\mu }{O}^{\mu }$ to ${⨁}_{\nu }\left(U_q^{\nu }\right(g_J\left)\text{-Mod}\right)$ using the canonical restriction functors. We then establish that $R_J$ is essentially surjective on finite-dimensional simple objects by constructing notable preimages that we call inflations.

We conjecture that all simple objects in $O_J^{sh}$ (which is the analog of $O^{sh}$ for the subalgebras $U_q^{\nu }\left(g_J\right)$) admit some inflation and prove this for $g$ of type A–B or $g_J$ a direct sum of copies of ${\mathrm{sl}}_2$ and ${\mathrm{sl}}_3$. We finally apply our results to deduce certain R-matrices and examples of cluster structures over Grothendieck rings.