We study the affine analogue $\mathrm{ FT}_p(\mathfrak{ sl}_2)$ of the triplet algebra. We show that $\mathrm{ FT}_p(\mathfrak{ sl}_2)$ is quasi-lisse and the associated variety is the nilpotent cone of $\mathfrak{ sl}_2$. We realize $\mathrm{ FT}_p(\mathfrak{ sl}_2)$ as the global sections of a sheaf of vertex algebras in the spirit of Feigin--Tipunin and thereby construct infinitely many simple modules and, in particular solve a conjecture by Semikhatov and Tipunin. We introduce the Kazama--Suzuki dual superalgebra $s\mathcal{ W}_p(\mathfrak{ sl}_{ 2|1})$ of $\mathrm{ FT}_p(\mathfrak{ sl}_2)$ and their singlet type subalgebras $s\mathcal{ M}_p(\mathfrak{ sl}_{ 2|1})$ and $\mathcal{ M}_p(\mathfrak{ sl}_2)$ and show their correspondence of categories. For $p=1$, we show the logarithmic Kazhdan--Lusztig correspondence for these (super)algebras and, in particular, show that the quantum group corresponding to $s\mathcal{ M}_1(\mathfrak{ sl}_{ 2|1})$ is the unrolled restricted quantum supergroup $u^H_{ -1}(\mathfrak{ sl}_{ 2|1})$ as suggested by Semikhatov and Tipunin.
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