We extend the notion of semi-infinite cohomology of Lie algebras to include cases where the Lie algebra does not admit a semi-infinite structure but satisfies a mild condition. We call the resulting cohomology the adjusted semi-infinite cohomology. We construct a square zero differential in the adjusted framework in a uniform way and give it a characterization. Our construction clarifies the definition of the affine W-algebra associated to a general nilpotent element given by V. Kac, S. Roan and M. Wakimoto.
q-Deformation of W(2,2) Lie algebra associated with quantum groups
Homogeneous Rota-Baxter operators on $3$-Lie algebra $A_ω$
General Formulas of the Structure Constants in the $\mathfrak{su}(N)$ Lie Algebra
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