Hopf-cyclic cohomology of bicrossed product Hopf algebras

TL;DR

This paper investigates the structure of SAYD modules over bicrossed product Hopf algebras derived from matched pairs of Lie groups and algebras, computing their Hopf-cyclic cohomology and establishing connections with Lie algebra cohomology.

Contribution

It classifies SAYD modules over Lie-Hopf algebras and links their cyclic cohomology to relative Lie algebra cohomology, providing new examples and a generalized framework.

Findings

Classification of SAYD modules over Lie-Hopf algebras.

Establishment of a van Est isomorphism for cyclic cohomology.

Construction of a nontrivial 4-dimensional SAYD module.

Abstract

In this dissertation we study the coefficients spaces (SAYD modules) of Hopf-cyclic cohomology theory over a certain family of bicrossed product Hopf algebras, and we compute the Hopf-cyclic cohomology of such Hopf algebras with coefficients. We associate a Hopf algebra, what we call a Lie-Hopf algebra, to any matched pair of Lie groups, Lie algebras and affine algebraic groups via the semi-dualization procedure of Majid. We then identify the SAYD modules over Lie-Hopf algebras with the representations and corepresentations of the total Lie group, Lie algebra or the affine algebraic group of the matched pair. First we classify the SAYD modules that correspond only to the representations of a total Lie group (algebra). We call them induced SAYD modules. We then generalize this identification, focusing on the matched pair of Lie algebras. We establish a one-to-one correspondence between the SAYD modules over the Lie-Hopf algebra associated to a matched pair of Lie algebras and certain SAYD modules over the total Lie algebra. Once the SAYD modules are associated to the representations and the corepresentations of Lie algebras, nontrivial examples can be constructed. This way, we illustrate a highly nontrivial 4-dimensional SAYD module over the Schwarzian Hopf algebra H_{1S}. In addition, we discuss the periodic cyclic cohomology of Lie-Hopf algebras with nontrivial SAYD coefficients. We obtain a general van Est isomorphism identifying the periodic cyclic cohomology of a Lie-Hopf algebra with the (relative) Lie algebra cohomology of the corresponding total Lie algebra.