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 $U_q^μ(\mathfrak{g})$ and the associated categories $\mathcal{O}^μ$ (defined by Hernandez). In particular, we introduce natural subalgebras $U_q^ν(\mathfrak{g}_J)\,{\subseteq}\,U_q^μ(\mathfrak{g})$ and obtain a functor $\mathcal{R}_J$ from $\mathcal{O}^{sh}\,{=}\bigoplus_μ\mathcal{O}^μ$ to $\bigoplus_ν(U_q^ν(\mathfrak{g}_J)\text{-Mod})$ 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 $\mathcal{O}^{sh}_J$ (which is the analog of $\mathcal{O}^{sh}$ for the subalgebras $U_q^ν(\mathfrak{g}_J)$) 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.
Shifted affine iquantum groups of quasi-split ADE types
Shenzhen International Center for Mathematics and Department of Mathematics, Southern University of Science and Technology（深圳国际数学中心及南方科技大学数学系）；Department of Mathematics, University of Virginia（弗吉尼亚大学数学系）；Département de mathématiques, Université de Sherbrooke（舍布鲁克大学数学系）
Shifted twisted Yangians and affine Grassmannian islices
Shenzhen International Center for Mathematics and Department of Mathematics, Southern University of Science and Technology（深圳国际数学中心及南方科技大学数学系）；Department of Mathematics, University of Virginia（弗吉尼亚大学数学系）；Département de mathématiques, Université de Sherbrooke（舍布鲁克大学数学系）
Notes on a homomorphism from the affine Yangian associated with $\hat{\mathfrak{sl}}(n)$ to the affine Yangian associated with $\widehat{\mathfrak{sl}}(n+1)$
Coulomb branches of $3d$ $\mathcal N=4$ quiver gauge theories and slices in the affine Grassmannian (with appendices by Alexander Braverman, Michael Finkelberg, Joel Kamnitzer, Ryosuke Kodera, Hiraku Nakajima, Ben Webster, and Alex Weekes)
