Batalin-Vilkovisky algebras and cyclic cohomology of Hopf algebras.

*(English)*Zbl 1101.19003One 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a BV structure on the Hochschild cohomology of \(A\) (with coefficients in \(A\)), giving back a result of Tradler [unpublished].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.

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.

Reviewer: Friedrich Wagemann (Nantes)

##### MSC:

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 |

##### Keywords:

Batalin-Vilkovisky algebra; cyclic cohomology; cyclic operad; Hopf algebra; Hochschild cohomology**OpenURL**

##### References:

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