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Multibraces on the Hochschild space. (English) Zbl 1008.17001

Summary: We generalize the coupled braces \(\{x\} \{y\}\) of Gerstenhaber and \(\{x\} \{y_1\dots,y_n\}\) of Gerstenhaber and Getzler depicting compositions of multilinear maps in the Hochschild space \(C^\bullet(A)= \operatorname{Hom}(T^\bullet A;A)\) of the graded vector space \(A\) to expressions of the form
\(\{x_1^{(1)},\dots, x_{i_1}^{(1)}\}\cdots \{x_1^{(m)},\dots, x_{i_m}^{(m)}\}\) on the extended space \(C^{\bullet,bullet}(A)= \operatorname{Hom}(T^\bullet A;T^\bullet A)\). We apply multibraces to study associative and Lie algebras, Batalin-Vilkovisky algebras, and \(A_\infty\) and \(L_\infty\) algebras: most importantly, we introduce a new variant of the master identity for \(L_\infty\) algebras in the form \(\{\widetilde{m} \circ \widetilde{m}\} \{sa_1\} \{sa_2\}\cdots \{sa_n\}= 0\). Using the new language, we also explain the significance of this notation for bialgebras (coassociativity is simply \(\Delta \circ \Delta=0\)), comment on the bialgebra cohomology differential of Gerstenhaber and Schack, and define multilinear higher-order differential operators with respect to multilinear maps.

MSC:

17A30 Nonassociative algebras satisfying other identities
17B55 Homological methods in Lie (super)algebras
18G55 Nonabelian homotopical algebra (MSC2010)
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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