Differential Geometry on Hopf Algebras and Quantum Groups (Ph.D. Thesis)

TL;DR

This thesis develops a framework for differential geometry on Hopf algebras and quantum groups, extending classical concepts to noncommutative settings and exploring their algebraic structures.

Contribution

It introduces a differential geometric approach to Hopf algebras, including quantum Lie algebras and Cartan calculus, using noncommutative geometry techniques.

Findings

Constructed differential geometry on Hopf algebras.

Defined quantum Lie algebras and their properties.

Derived Cartan calculus for quantum groups.

Abstract

The differential geometry on a Hopf algebra is constructed, by using the basic axioms of Hopf algebras and noncommutative differential geometry. The space of generalized derivations on a Hopf algebra of functions is presented via the smash product, and used to define and discuss quantum Lie algebras and their properties. The Cartan calculus of the exterior derivative, Lie derivative, and inner derivation is found for both the universal and general differential calculi of an arbitrary Hopf algebra, and, by restricting to the quasitriangular case and using the numerical R-matrix formalism, the aforementioned structures for quantum groups are determined.