Inflations for representations of shifted quantum affine algebras

Fix a finite-dimensional simple Lie algebra g and let g J subset of 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 (g) and the associated categories O mu (defined by Hernandez). In particular, we introduce natural subalgebras U q nu (gJ) subset of U q mu ( g ) and obtain a functor RJ from O sh =(R)mu O mu to (R) nu ( U q nu (gJ)-Mod) using the canonical restriction functors. We then establish that RJ is essentially surjective on finite-dimensional simple objects by constructing notable preimages that we call inflations. We conjecture that all simple objects in O sh J (which is the analog of O sh for the subalgebras U q nu (gJ)) admit some inflation and prove this for g of type A-B or g J a direct sum of copies of sl2 and sl3. We finally apply our results to deduce certain R-matrices and examples of cluster structures over Grothendieck rings. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).