Braided $$\varvec{L_{\infty }}$$-algebras, braided field theory and noncommutative gravity

We define a new homotopy algebraic structure, that we call braided $L_\infty$-algebra, and use it to systematically construct class of noncommutative field theories, theories. Braided theories have gauge symmetries which realize Lie algebra, whose Noether identities are inhomogeneous extensions the classical identities, do not act on solutions equations. Drinfel'd twist deformation quantization techniques generate deformations with symmetries, compare more conventional star-gauge symmetries. apply our formalism introduce version general relativity without matter fields in Einstein-Cartan-Palatini formalism. In limit vanishing parameter, theory gravity reduces any extensions.