Quantum groups, noncommutative differential geometry and applications
Abstract
The topic of this thesis is the development of a versatile and geometrically motivated differential calculus on noncommutative or quantum spaces, providing powerful but easytouse mathematical tools for applications in physics and related sciences. A generalization of unitary time evolution is proposed and studied for a simple 2level system, leading to nonconservation of microscopic entropy, a phenomenon new to quantum mechanics. A Cartan calculus that combines functions, forms, Lie derivatives and inner derivations along general vector fields into one big algebra is constructed for quantum groups and then extended to quantum planes. The construction of a tangent bundle on a quantum group manifold and an BRST type approach to quantum group gauge theory are given as further examples of applications. The material is organized in two parts: Part I studies vector fields on quantum groups, emphasizing Hopf algebraic structures, but also introducing a ``quantum geometric`` construction. Using a generalized semidirect product construction we combine the dual Hopf algebras A of functions and U of leftinvariant vector fields into one fully bicovariant algebra of differential operators. The pure braid group is introduced as the commutant of {Delta}(U). It provides invariant maps A {yields} U and thereby bicovariant vector fields, casimirs andmore »
 Authors:

 Lawrence Berkeley Lab., CA (United States); California Univ., Berkeley, CA (United States). Dept. of Physics
 Publication Date:
 Research Org.:
 Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
 Sponsoring Org.:
 USDOE; National Science Foundation (NSF)
 OSTI Identifier:
 10148553
 Report Number(s):
 LBL34942; UCBPTH93/35
ON: DE94011358; CNN: PHY9021139
 DOE Contract Number:
 AC0376SF00098
 Resource Type:
 Thesis/Dissertation
 Resource Relation:
 Related Information: Thesis (Ph.D.)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; LIE GROUPS; QUANTUM MECHANICS; DIFFERENTIAL GEOMETRY; ALGEBRA; 661100; 990200; CLASSICAL AND QUANTUM MECHANICS; MATHEMATICS AND COMPUTERS
Citation Formats
Schupp, Peter. Quantum groups, noncommutative differential geometry and applications. United States: N. p., 1993.
Web. doi:10.2172/10148553.
Schupp, Peter. Quantum groups, noncommutative differential geometry and applications. United States. doi:10.2172/10148553.
Schupp, Peter. Thu .
"Quantum groups, noncommutative differential geometry and applications". United States. doi:10.2172/10148553. https://www.osti.gov/servlets/purl/10148553.
@article{osti_10148553,
title = {Quantum groups, noncommutative differential geometry and applications},
author = {Schupp, Peter},
abstractNote = {The topic of this thesis is the development of a versatile and geometrically motivated differential calculus on noncommutative or quantum spaces, providing powerful but easytouse mathematical tools for applications in physics and related sciences. A generalization of unitary time evolution is proposed and studied for a simple 2level system, leading to nonconservation of microscopic entropy, a phenomenon new to quantum mechanics. A Cartan calculus that combines functions, forms, Lie derivatives and inner derivations along general vector fields into one big algebra is constructed for quantum groups and then extended to quantum planes. The construction of a tangent bundle on a quantum group manifold and an BRST type approach to quantum group gauge theory are given as further examples of applications. The material is organized in two parts: Part I studies vector fields on quantum groups, emphasizing Hopf algebraic structures, but also introducing a ``quantum geometric`` construction. Using a generalized semidirect product construction we combine the dual Hopf algebras A of functions and U of leftinvariant vector fields into one fully bicovariant algebra of differential operators. The pure braid group is introduced as the commutant of {Delta}(U). It provides invariant maps A {yields} U and thereby bicovariant vector fields, casimirs and metrics. This construction allows the translation of undeformed matrix expressions into their less obvious quantum algebraic counter parts. We study this in detail for quasitriangular Hopf algebras, giving the determinant and orthogonality relation for the ``reflection`` matrix. Part II considers the additional structures of differential forms and finitely generated quantum Lie algebras  it is devoted to the construction of the Cartan calculus, based on an undeformed Cartan identity.},
doi = {10.2172/10148553},
journal = {},
number = ,
volume = ,
place = {United States},
year = {1993},
month = {12}
}