Twisted Root Decompositions and Brst Cohomology in Quantum Reduction

We study twisted root decompositions of affine Lie algebras, obtained by interchanging the roles of some positive and negative roots. Such decompositions arise naturally in cases such as generalized Verma modules, nonregular Whittaker modules and related settings. To study these mixed root decompositions within a categorical framework, we introduce a new category that provides a natural homological setting for BRST reduction. In our new category, we prove that the BRST cohomology of every object vanishes in all nonzero degrees. This establishes the exactness of the BRST functor and demonstrates that the proposed category offers an effective environment for studying twisted root decompositions.