The vertex algebras $V^{ (p)}$ and $R^{ (p)}$ introduced in [2] are very interesting relatives of the famous triplet algebras of logarithmic CFT. The algebra $V^{ (p)}$ (respectively, $R^{ (p)}$) is a large extension of the simple affine vertex algebra $L_k(\mathfrak{ sl}_2)$ (respectively, $L_k(\mathfrak{ sl}_2)$ times a Heisenberg algebra), at level $k=-2+1/p$ for positive integer $p$. In this paper, we derive structural results of these algebras and prove various conjectures coming from representation theory and physics. We show that SU(2) acts as automorphisms on $V^{ (p)}$ and we decompose $V^{ (p)}$ as an $L_k(\mathfrak{ sl}_2)$-module and $R^{ (p)}$ as an $L_k(\mathfrak{ gl}_2)$-module. The decomposition of $V^{ (p)}$ shows that $V^{ (p)}$ is the large level limit of a corner vertex algebra appearing in the context of S-duality. We also show that the quantum Hamiltonian reduction of $V^{ (p)}$ is the logarithmic doublet algebra $A^{ (p)}$ introduced in [12], while the reduction of $R^{ (p)}$ yields the $B^{ (p)}$-algebra of [39]. Conversely, we realize $V^{ (p)}$ and $R^{ (p)}$ from $A^{ (p)}$ and $B^{ (p)}$ via a procedure that deserves to be called inverse quantum Hamiltonian reduction. As a corollary, we obtain that the category $KL_{ k}$ of ordinary $L_k(\mathfrak{ sl}_2)$-modules at level $k=-2+1/p$ is a rigid vertex tensor category equivalent to a twist of the category Rep$(SU(2))$. This finally completes rigid braided tensor category structures for $L_k(\mathfrak{ sl}_2)$ at all levels $k$. We also establish a uniqueness result of certain vertex operator algebra extensions and use this result to prove that both $R^{ (p)}$ and $B^{ (p)}$ are certain non-principal W-algebras of type $A$ at boundary admissible levels. The same uniqueness result also shows that $R^{ (p)}$ and $B^{ (p)}$ are the chiral algebras of Argyres-Douglas theories of type $(A_1, D_{ 2p})$ and $(A_1, A_{ 2p-3})$.
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