  The notion of shifted quantum groups has recently played an important role in
algebraic geometry. This subtle modification of the original definition brings
more flexibility in the representation theory of quantum groups. The first part
of this paper presents new mathematical results for the shifted quantum
toroidal $\mathfrak{gl}(1)$ and quantum affine $\mathfrak{sl}(2)$ algebras
(resp. denoted $\ddot{U}_{q_1,q_2}^\boldsymbol{\mu}(\mathfrak{gl}(1))$ and
$\dot{U}_q^\boldsymbol{\mu}(\mathfrak{sl}(2))$). It defines several new
representations, including finite dimensional highest $\ell$-weight
representations for the toroidal algebra, and a vertex representation of
$\dot{U}_q^\boldsymbol{\mu}(\mathfrak{sl}(2))$ acting on Hall-Littlewood
polynomials. It also explores the relations between representations of
$\dot{U}_q^\boldsymbol{\mu}(\mathfrak{sl}(2))$ and
$\ddot{U}_{q_1,q_2}^\boldsymbol{\mu}(\mathfrak{gl}(1))$ in the limit
$q_1\to\infty$ ($q_2$ fixed), and present the construction of several new
intertwiners. These results are used in the second part to construct BPS
observables for 5d $\mathcal{N}=1$ and 3d $\mathcal{N}=2$ gauge theories. In
particular, it is shown that 5d hypermultiplets and 3d chiral multiplets can be
introduced in the algebraic engineering framework using shifted
representations, and the Higgsing procedure is revisited from this perspective.
