Inflations for representations of shifted quantum affine algebras

Abstract

Fix a finite-dimensional simple Lie algebra $\mathfrak{g}$ and let ${\mathfrak{g}}_J\subseteq \mathfrak{g}$ be a Lie subalgebra coming from a Dynkin diagram inclusion. Then, the corresponding restriction functor is not essentially surjective on finite-dimensional simple ${\mathfrak{g}}_J$-modules. In this article, we study Finkelberg-Tsymbaliuk's shifted quantum affine algebras $U_q^{\mu }(\mathfrak{g})$ and the associated categories ${\mathcal{O}}^{\mu }$ (defined by Hernandez). In particular, we introduce natural subalgebras $U_q^{\nu }({\mathfrak{g}}_J)\subseteq U_q^{\mu }(\mathfrak{g})$ and obtain a functor ${\mathcal{R}}_J$ from ${\mathcal{O}}^{sh}={⨁}_{\mu }{\mathcal{O}}^{\mu }$ to ${⨁}_{\nu }(U_q^{\nu }({\mathfrak{g}}_J)\text{-Mod})$ using the canonical restriction functors. We then establish that ${\mathcal{R}}_J$ is essentially surjective on finite-dimensional simple objects by constructing notable preimages that we call inflations. We conjecture that all simple objects in ${\mathcal{O}}_J^{sh}$ (which is the analog of ${\mathcal{O}}^{sh}$ for the subalgebras $U_q^{\nu }({\mathfrak{g}}_J)$) admit some inflation and prove this for $\mathfrak{g}$ of type A-B or ${\mathfrak{g}}_J$ a direct sum of copies of ${\mathfrak{sl}}_2$ and ${\mathfrak{sl}}_3$. We finally apply our results to deduce certain $R$-matrices and examples of cluster structures over Grothendieck rings.

arXiv comment37 pages, comments welcome

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