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Title: Quantum groups, non-commutative differential geometry and applications

Abstract

The topic of this thesis is the development of a versatile and geometrically motivated differential calculus on non-commutative or quantum spaces, providing powerful but easy-to-use mathematical tools for applications in physics and related sciences. A generalization of unitary time evolution is proposed and studied for a simple 2-level system, leading to non-conservation 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 semi-direct product construction we combine the dual Hopf algebras A of functions and U of left-invariant 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 » 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.« less

Authors:
 [1]
  1. 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):
LBL-34942; UCB-PTH-93/35
ON: DE94011358; CNN: PHY-90-21139
DOE Contract Number:  
AC03-76SF00098
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, non-commutative differential geometry and applications. United States: N. p., 1993. Web. doi:10.2172/10148553.
Schupp, Peter. Quantum groups, non-commutative differential geometry and applications. United States. doi:10.2172/10148553.
Schupp, Peter. Thu . "Quantum groups, non-commutative differential geometry and applications". United States. doi:10.2172/10148553. https://www.osti.gov/servlets/purl/10148553.
@article{osti_10148553,
title = {Quantum groups, non-commutative 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 non-commutative or quantum spaces, providing powerful but easy-to-use mathematical tools for applications in physics and related sciences. A generalization of unitary time evolution is proposed and studied for a simple 2-level system, leading to non-conservation 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 semi-direct product construction we combine the dual Hopf algebras A of functions and U of left-invariant 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}
}

Thesis/Dissertation:
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