Multibraces on the Hochschild space

We generalize the coupled braces {x}{y} of Gerstenhaber and {x}{y1,…,yn} of Gerstenhaber and Getzler depicting compositions of multilinear maps in the Hochschild space C•(A)=Hom(T•A;A) of a graded vector space A to expressions of the form {x1(1),…,xi1(1)}cdots, three dots, centered{x1(m),…,xim(m)} on the extended space C•,•(A)=Hom(T•A;T•A). We apply multibraces to study associative and Lie algebras, Batalin–Vilkovisky algebras, and A∞ and L∞ algebras: most importantly, we introduce a new variant of the master identity for L∞ algebras in the form image. Using the new language, we also explain the significance of this notation for bialgebras (coassociativity is simply Δring operatorΔ=0), comment on the bialgebra cohomology differential of Gerstenhaber and Schack, and define multilinear higher-order differential operators with respect to multilinear maps.