Shifted quantum affine symmetric pairs: finite presentations and graded coideal structures

Shifted quantum affine symmetric pair algebras were recently introduced as a trigonometric analogue of shifted twisted Yangians, whose truncations are expected to quantize fixed point loci in generalized affine Grassmannian slices. They are defined via a loop presentation with infinitely many generators and relations. In this paper, we consider anti-dominantly shifted quantum affine symmetric pair algebras of simply-laced split type. Our main result is a construction of a finite presentation à la Levendorskii. We apply it in type $\mathsf{A}$ to construct an embedding of a shifted quantum symmetric pair algebra into the corresponding shifted quantum affine algebra, yielding a graded coideal structure.