Quantum Groups and Asymptotic Symmetries

TL;DR

This thesis explores the algebraic structures underlying asymptotic spacetime symmetries, classifies related Lie bialgebras, constructs Hopf algebras, and investigates their phenomenological and theoretical implications.

Contribution

It provides new classifications of Lie bialgebras associated with asymptotic symmetries and analyzes their quantum group structures and physical consequences.

Findings

Classification of Lie bialgebras related to asymptotic symmetries

Construction and analysis of specific Hopf algebras

Implications for quantum gravity phenomenology and black hole information

Abstract

This thesis is devoted to the study of Lie bialgebra and Hopf algebra structures related to certain versions of non-commutative geometry constructed on infinite-dimensional Lie algebras that arise in the context of asymptotic symmetries of spacetime. We prove a number of theorems about cohomology groups that aid the classification of the Lie bialgebras and explicitly construct and analyze selected Hopf algebras. Particularly interesting behavior was found by studying the contraction limit of spacetimes with cosmological constant and the inclusion of central charges on the level of Lie bialgebras and Hopf algebras. Phenomenological consequences, like deformed in-vacuo dispersion relations, known from the study of $\kappa$-Poincar\'e quantum groups, are investigated. Furthermore, we examine how a new proposal in the context of the black hole information loss paradox and counterarguments against it are affected.