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Pré-Publication, Document De Travail Année : 2024

Inflations for representations of shifted quantum affine algebras

Résumé

Fix a finite-dimensional simple Lie algebra $\mathfrak{g}$ and let $\mathfrak{g}_J\subseteq \mathfrak{g}$ be a non-trivial Lie subalgebra coming from an inclusion of Dynkin diagrams. Then, the restriction functor from $\mathfrak{g}$ to $\mathfrak{g}_J$ is not essentially surjective on finite-dimensional simple $\mathfrak{g}_J$-modules. In this article, we study the shifted quantum affine algebra $U_q^{\mu}(\mathfrak{g})$ of Finkelberg--Tsymbaliuk and the category $\mathcal{O}^{\mu}$ associated to this algebra by Hernandez (for $\mu\in \Lambda^{\vee}$ a coweight of $\mathfrak{g}$). More precisely, we consider the restriction functors from $U_q^{\mu}(\mathfrak{g})$ to natural subalgebras $U_q^{\nu}(\mathfrak{g}_J)$ (with $\nu$ the projection of $\mu$ on the coweight lattice $\Lambda^{\vee}_J$ of $\mathfrak{g}_J$) and use this data to obtain a functor from $\mathcal{O}^{sh}=\bigoplus_{\mu\in\Lambda^{\vee}} \mathcal{O}^{\mu}$ to $\bigoplus_{\nu\in \Lambda^{\vee}_J} (U_q^{\nu}(\mathfrak{g}_J)\text{-Mod})$. We then establish that the resulting functor is essentially surjective on finite-dimensional simple objects by constructing notable preimages for it, called inflations. We conjecture that all simple, possibly infinite-dimensional, modules in $\mathcal{O}^{sh}_J$ (which is the equivalent of $\mathcal{O}^{sh}$ for the subalgebras $U_q^{\nu}(\mathfrak{g}_J)$) admit some inflation and prove this conjecture if either $\mathfrak{g}$ is of type A-B or if $\mathfrak{g}_J$ is isomorphic (as a Lie algebra) to a direct sum of copies of $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$. We also give applications of our results to the study of compatible cluster structures over Grothendieck rings and to the computation of particular $R$-matrices in $\mathcal{O}^{sh}$.

Dates et versions

hal-04533110 , version 1 (04-04-2024)

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Théo Pinet. Inflations for representations of shifted quantum affine algebras. 2024. ⟨hal-04533110⟩
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