Kowalzig, Niels Batalin-Vilkovisky algebra structures on (Co)Tor and Poisson bialgebroids. (English) Zbl 1350.16012 J. Pure Appl. Algebra 219, No. 9, 3781-3822 (2015). Summary: In this article, we extend our preceding studies on higher algebraic structures of (co)homology theories defined by a left bialgebroid \((U,A)\). For a braided commutative Yetter-Drinfel’d algebra \(N\), explicit expressions for the canonical Gerstenhaber algebra structure on \(\mathrm{Ext}_U(A,N)\) are given. Similarly, if \((U,A)\) is a left Hopf algebroid where \(A\) is an anti-Yetter-Drinfel’d module over \(U\), it is shown that the cochain complex computing \(\mathrm{Cotor}_U(A,N)\) defines a cyclic operad with multiplication and hence the groups \(\mathrm{Cotor}_U(A,N)\) form a Batalin-Vilkovisky algebra. In the second part of this article, Poisson structures and the Poisson bicomplex for bialgebroids are introduced, which simultaneously generalise, for example, classical Poisson as well as cyclic homology. In case the bialgebroid \(U\) is commutative, a Poisson structure on \(U\) leads to a Batalin-Vilkovisky algebra structure on \(\mathrm{Tor}_U(A,A)\). As an illustration, we show how this generalises the classical Koszul bracket on differential forms, and conclude by indicating how classical Lie-Rinehart bialgebras (or, geometrically, Lie bialgebroids) arise from left bialgebroids. Cited in 1 ReviewCited in 6 Documents MSC: 16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) 16T05 Hopf algebras and their applications 16T10 Bialgebras 19D55 \(K\)-theory and homology; cyclic homology and cohomology 58B34 Noncommutative geometry (à la Connes) Keywords:cohomology theories; homology theories; left bialgebroids; Lie-Rinehart bialgebras; braided commutative Yetter-Drinfel’d algebras; Gerstenhaber algebras; cochain complexes; cyclic homology; Poisson homology PDFBibTeX XMLCite \textit{N. Kowalzig}, J. Pure Appl. Algebra 219, No. 9, 3781--3822 (2015; Zbl 1350.16012) Full Text: DOI arXiv References: [1] Abraham, R.; Marsden, J., Foundations of Mechanics (1967), W.A. Benjamin, Inc.: W.A. Benjamin, Inc. 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