Various doublings of Hopf Algebras. Algebras of Operators on Quantum Groups and Complex Cobordisms

TL;DR

This paper introduces new algebraic structures called doublings of Hopf algebras, which serve as quantum analogs of differential operator rings on groups and have significant applications in topology and analysis.

Contribution

It constructs and studies a family of doubled Hopf algebra structures that generalize quantum doubles and exhibit important 'almost Hopf' properties.

Findings

Constructed a family of doubled Hopf algebras based on algebra and dual product.

Identified these structures as quantum analogs of differential operator rings.

Demonstrated their relevance in topological and analytical contexts.

Abstract

Family of doublings of Hopf algeras based on the product of algebra and its dual are constructed and studied. Special cases of these construction may be considered as natural quantum analogs of rings of differential operators on groups. Such constructions appeared in 1966-67 in the authors works in the complex cobordism theory. These constructions do not leed to the Hopf algebras (except of the special case of the Drinfeld's quantum double which is not the same as the natural quantum analog of the ring of operators on groups). However, they have important ''almost Hopf'' properties important in particular in the topological and analytical applications.