Cocyclic complexes of Hopf algebras with special antipodes

Abstract

We show that for any bimodule M and bicomodule C of a Hopf algebra H with involutive antipode, Hochschild complex $C^{*}(H,M)$ and $C^{*}(C,H)$ are cocyclic $\mathbb{K}$-modules. Their para-cocyclic operators are compatible in Gerstenhaber-Schack bicomplex of a Hopf algebra morphism $\phi :A\stackrel{}{\to }B,$ which is a combination of Hochschild complex $C^{*}(A,B^{\otimes q})$ and coHochschild complex $C^{*}(A^{\otimes p},B)$ for any $p,q\ge 0,$ and make the bicomplex into a cylindrical $\mathbb{K}$-module under appropriate conditions. From this bicomplex, we get a cocyclic structure of the diagonal complex. With these cocyclic operators, the operadic structure of the diagonal complex is cyclic, so that the Gerstenhaber algebra structure on the diagonal cohomology evolves into a Batalin-Vilkovisky algebra. Hence we obtain the structure of Gerstenhaber-Schack cohomology of $\phi ,$ which is isomorphic to the diagonal cohomology. Our results can be applied to group algebra $\mathbb{K}G$ and arbitrary G-graded Hopf algebra A. Let $\phi$ be ϵ_A (resp. u_B, id_A). Then we can get the Gerstenhaber structure on Hochschild cohomology $C^{*}(A,\mathbb{K})$ (resp. Adams Cobar construction of B, Gerstenhaber-Schack cohomology of A). Moreover, the Gerstenhaber structure operators of ϵ_A and u_B commute with composition.

Keywords:

• Batalin-Vilkovisky algebras
• bialgebra pair
• cocyclic homology
• Gerstenhaber algebras
• Gerstenhaber-Schack cohomology

2020 Mathematics Subject Classification:

• 16E40
• 18D50
• 18G60
• 19D55
• 16T05

Acknowledgments

The authors would like to thank the referee for their careful reading and helpful comments on this article. This article is greatly improved by their illuminating suggestion.

Additional information

Funding

The authors are sponsored by NNSFC (No.11871421, 12171129), ZJNSF (No. LY17A010015).