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Batalin-Vilkovisky algebras and cyclic cohomology of Hopf algebras. (English) Zbl 1101.19003
One of the central ideas of homotopy theory is that additional algebraic structure on the (co)homology or homotopy of a space may be helpful in computing this (co)homology or homotopy. From this point of view, it is an interesting remark that some structures of a Gerstenhaber bracket may be derived from a Batalin-Vilkovisky (BV) structure. Here a (graded) Gerstenhaber structure on a graded vector space is a graded product and a bracket of degree \(-1\) which satisfy a Leibniz identity. The main example is the Gerstenhaber structure on the Hochschild cohomology of an associative algebra, found by M. Gerstenhaber [Ann. Math. (2) 78, 267–288 (1963; Zbl 0131.27302)]. On the other hand, a BV structure on a graded vector space is a graded product and a degree \(-1\) operator \(B\) such that \(B^2=0\) and which satisfies a rather complicated identity. Setting \[ \{a,b\}:=(-1)^{| a| }(B(ab)-B(a)b-(-1)^{| a| }aB(b)), \] one recovers from a BV structure a Gerstenhaber structure.
The article under review shows that the Adams cobar construction \(\Omega{\mathcal H}\) on a Hopf algebra \({\mathcal H}\) carries a BV structure on its cohomology which is derived from the cocyclic module structure of A. Connes and H. Moscovici [Commun. Math. Phys. 198, 199–246 (1998; Zbl 0940.58005)] and which gives back the classical Gerstenhaber structure.
The proof relies on the operadic description of the involved structures, and the author shows in particular that for a cyclic operad \(O\) with multiplication, the structure of a cosimplicial module extends to a structure of a cocyclic module such that Connes boundary operator on the associated cochain complex induces a BV structure.
Another context where similar constructions are possible is the theory of symmetric algebras, i.e., an algebra \(A\) equipped with an isomorphism of \(A\)-bimodules \(A\cong A^*\) of \(A\) with its linear dual. As a corollary, the author obtains a BV structure on the Hochschild cohomology of \(A\) (with coefficients in \(A\)), giving back a result of Tradler [unpublished].
In related work, Kaufmann, Tradler-Zeinalian [T. Tradler and M. Zeinalian, J. Pure Appl. Algebra 204, 280–299 (2006; Zbl 1147.16012)], Costello and Kontsevich-Soibelman prove that the BV algebra structure on \(HH^{*}(A,A)\) comes from the action of various operads or PROPs on the Hochschild cochain complex \(\mathcal{C}^{*}(A,A)\): the so-called cyclic Deligne conjecture.

19D55 \(K\)-theory and homology; cyclic homology and cohomology
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
18D50 Operads (MSC2010)
55P48 Loop space machines and operads in algebraic topology
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
17B70 Graded Lie (super)algebras
17B62 Lie bialgebras; Lie coalgebras
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