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.


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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