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We summarise recent perspectives on symmetries of noncommutative fieldtheories based on homotopy algebras. We show how these viewpoints naturallylead to a new class of noncommutative field theories which possess braidedgauge symmetries, and explain in detail their uses in gravity. We review howthese considerations lead to a new theory of noncommutative gravity in fourdimensions within the Einstein-Cartan-Palatini formalism.

The $L_{\infty}$-structure of noncommutative gravity