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Quasi-Lie Algebras and Quasi-Deformations. Algebraic Structures Associated with Twisted Derivations

Larsson, Daniel LU (2006)
Abstract
This thesis introduces a new deformation scheme for Lie algebras, which we refer to as ?quasi-deformations? to clearly distinguish it from the classical Grothendieck-Schlessinger and Gers-tenhaber deformation schemes. The main difference is that quasi-deformations are not in gene-ral category-preserving, i.e., quasi-deforming a Lie algebra gives an object in the larger catego-ry of ?quasi-Lie algebras?, a notion which is also introduced in this thesis. The quasi-deforma-tion scheme can be loosely described as follows: represent a Lie algebra by derivations acting on a commutative, associative algebra with unity and replace these derivations with twisted versions. An algebra structure is then imposed, thus arriving at the quasi-deformed... (More)
This thesis introduces a new deformation scheme for Lie algebras, which we refer to as ?quasi-deformations? to clearly distinguish it from the classical Grothendieck-Schlessinger and Gers-tenhaber deformation schemes. The main difference is that quasi-deformations are not in gene-ral category-preserving, i.e., quasi-deforming a Lie algebra gives an object in the larger catego-ry of ?quasi-Lie algebras?, a notion which is also introduced in this thesis. The quasi-deforma-tion scheme can be loosely described as follows: represent a Lie algebra by derivations acting on a commutative, associative algebra with unity and replace these derivations with twisted versions. An algebra structure is then imposed, thus arriving at the quasi-deformed algebra. Therefore the quasi-deformation takes place on the level of representations: we ?deform? the representation, which is then ?pulled-back? to an algebra structure.



The different Chapters of this thesis is concerned with different aspects of this quasi-deforma-tion scheme, for instance: Burchnall-Chaundy theory for the q-deformed Heisenberg algebra (Chapter II), the (quasi-Lie) algebraic structure on the vector space of twisted derivations (Chapter III), deformed Witt, Virasoro and loop algebras (Chapter III and IV), Central exten-sion theory (Chapter III and IV), the Lie algebra sl(2) and some associated quadratic algebras. (Less)
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author
supervisor
opponent
  • Professor Fuchs, Jürgen, Institutionen för ingenjörsvetenskap, fysik och matematik, Karlstad Universitet
organization
publishing date
type
Thesis
publication status
published
subject
keywords
Talteori, fältteori, algebraisk geometri, algebra, gruppteori, Lie Algebras, Quasi-Deformations, Number Theory, field theory, algebraic geometry, group theory, Quasi-Lie Algebras
pages
157 pages
publisher
Department of Mathematics, Lund University
defense location
Room MH:C, Center for Mathematical Sciences, Sölvegatan 18, Lund Institute of Technology
defense date
2006-02-24 13:15
ISBN
91-628-6739-3
language
English
LU publication?
yes
id
8d2ceb6e-f034-4d0c-bf88-f916101e3bd0 (old id 546239)
date added to LUP
2007-09-27 15:45:00
date last changed
2016-09-19 08:45:07
@misc{8d2ceb6e-f034-4d0c-bf88-f916101e3bd0,
  abstract     = {This thesis introduces a new deformation scheme for Lie algebras, which we refer to as ?quasi-deformations? to clearly distinguish it from the classical Grothendieck-Schlessinger and Gers-tenhaber deformation schemes. The main difference is that quasi-deformations are not in gene-ral category-preserving, i.e., quasi-deforming a Lie algebra gives an object in the larger catego-ry of ?quasi-Lie algebras?, a notion which is also introduced in this thesis. The quasi-deforma-tion scheme can be loosely described as follows: represent a Lie algebra by derivations acting on a commutative, associative algebra with unity and replace these derivations with twisted versions. An algebra structure is then imposed, thus arriving at the quasi-deformed algebra. Therefore the quasi-deformation takes place on the level of representations: we ?deform? the representation, which is then ?pulled-back? to an algebra structure.<br/><br>
<br/><br>
The different Chapters of this thesis is concerned with different aspects of this quasi-deforma-tion scheme, for instance: Burchnall-Chaundy theory for the q-deformed Heisenberg algebra (Chapter II), the (quasi-Lie) algebraic structure on the vector space of twisted derivations (Chapter III), deformed Witt, Virasoro and loop algebras (Chapter III and IV), Central exten-sion theory (Chapter III and IV), the Lie algebra sl(2) and some associated quadratic algebras.},
  author       = {Larsson, Daniel},
  isbn         = {91-628-6739-3},
  keyword      = {Talteori,fältteori,algebraisk geometri,algebra,gruppteori,Lie Algebras,Quasi-Deformations,Number Theory,field theory,algebraic geometry,group theory,Quasi-Lie Algebras},
  language     = {eng},
  pages        = {157},
  publisher    = {ARRAY(0x97b1ee0)},
  title        = {Quasi-Lie Algebras and Quasi-Deformations. Algebraic Structures Associated with Twisted Derivations},
  year         = {2006},
}