We study locally finitary realizations of simple transitive module categories of infinite rank over the monoidal category $\mathscr {C}$ of finite-dimensional modules for the complex Lie algebra $\mathfrak {sl}_2$. Combinatorics of such realizations is governed by six infinite Coxeter diagrams. We show that five of these are realizable in our setup, while one (type $B_\infty $) is not. We also describe the $\mathscr {C}$-module subcategories of $\mathfrak {sl}_2$-mod generated by simple modules as well as the $\mathscr {C}$-module categories coming from the natural action of $\mathscr {C}$ on the categories of finite-dimensional modules over Lie subalgebras of $\mathfrak {sl}_2$.

An equivariant map queer Lie superalgebra is the Lie superalgebra of regular maps from an algebraic variety (or scheme) $X$ to a queer Lie superalgebra $\mathfrak{q}$ that are equivariant with respect to the action of a finite group $\Gamma $ acting on $X$ and $\mathfrak{q}$ . In this paper, we classify all irreducible finite-dimensional representations of the equivariant map queer Lie superalgebras under the assumption that $\Gamma $ is abelian and acts freely on $X$ . We show that such representations are parameterized by a certain set of $\Gamma $ -equivariant finitely supported maps from $X$ to the set of isomorphism classes of irreducible finite-dimensional representations of $\mathfrak{q}$ . In the special case where $X$ is the torus, we obtain a classification of the irreducible finite-dimensional representations of the twisted loop queer superalgebra.