Higher Homotopy Hopf Algebras Found: A Ten Year Retrospective

TL;DR

This paper reviews the decade-long search and development of higher homotopy Hopf algebras, highlighting their theoretical evolution and the discovery of A_infty-bialgebras.

Contribution

It provides a retrospective on the conceptual development and formalization of higher homotopy Hopf algebras over ten years.

Findings

Higher homotopy Hopf algebras are formally characterized as A_infty-bialgebras.

Deformation theory played a key role in understanding these structures.

The discovery of A_infty-bialgebras was motivated by limitations in existing cohomology detection.

Abstract

The search for higher homotopy Hopf algebras (known today as A_\infty-bialgebras) began in 1996 during a conference at Vassar College honoring Jim Stasheff in the year of his 60th birthday. In a talk entitled "In Search of Higher Homotopy Hopf Algebras", I indicated that a DG Hopf algebra could be thought of as some (unknown) higher homotopy structure with trivial higher order structure and deformed using a graded version of Gerstenhaber and Schack's bialgebra deformation theory. In retrospect, the bi(co)module structure encoded in Gerstenhaber and Schack's differential defining deformation cohomology detects some (but not all) of the A_infty-bialgebra structure relations. Nevertheless, this motivated the discovery of A_infty-bialgebras by S. Saneblidze and myself in 2005.