We apply the modern Batalin-Vilkovisky quantization techniques of Costello and Gwilliam to noncommutative field theories in finite-dimensional case fuzzy spaces. further develop a generalization this framework that are equivariant under triangular Hopf algebra symmetry, which particular leads quantizations analogs proposed recently through notion `braided $L_\infty$-algebras'. The illustrated b...

An explicit, geometric description of the first-class constraints and their Poisson brackets for gravity in Palatini-Cartan formalism (in space-time dimension greater than three) is given. The corresponding Batalin- Fradkin-Vilkovisky (BFV) formulation also developed.