A Lie algebraic pattern behind logarithmic CFTs

We introduce a purely Lie algebraic formalization of the Feigin--Tipunin's geometric construction of logarithmic CFTs/VOAs. After reformulating the geometric representation theory of FT construction under this new setting, within this framework, we uniformly construct the (multiplet) principal W-algebras at positive integer level associated with any simple Lie algebra $\mathfrak{g}$ and Lie superalgebra $\mathfrak{osp}(1|2r)$, thereby establishing Weyl-type character formulas and simplicity theorems that extend the second author's previous results.