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Algebraic Dependence of Commuting Elements in Algebras

Silvestrov, Sergei LU ; Svensson, Charlotte LU and de Jeu, M. (2009) International Workshop of Baltic-Nordic Algebra, Geometry and Mathematical Physics In Generalized Lie Theory in Mathematics, Physics and Beyond p.265-280
Abstract
The aim of this paper to draw attention to several aspects of the algebraic dependence in algebras. The article starts with discussions of the algebraic dependence problem in commutative algebras. Then the Burchnall-Chaundy construction for proving algebraic dependence and obtaining the corresponding algebraic curves for commuting differential operators in the Heisenberg algebra is reviewed. Next some old and new results on algebraic dependence of commuting q-difference operators and elements in q-deformed Heisenberg algebras are reviewed. The main ideas and essence of two proofs of this are reviewed and compared. One is the algorithmic dimension growth existence proof. The other is the recent proof extending the Burchnall-Chaundy approach... (More)
The aim of this paper to draw attention to several aspects of the algebraic dependence in algebras. The article starts with discussions of the algebraic dependence problem in commutative algebras. Then the Burchnall-Chaundy construction for proving algebraic dependence and obtaining the corresponding algebraic curves for commuting differential operators in the Heisenberg algebra is reviewed. Next some old and new results on algebraic dependence of commuting q-difference operators and elements in q-deformed Heisenberg algebras are reviewed. The main ideas and essence of two proofs of this are reviewed and compared. One is the algorithmic dimension growth existence proof. The other is the recent proof extending the Burchnall-Chaundy approach from differential operators and the Heisenberg algebra to the q-deformed Heisenberg algebra, showing that the Burchnall-Chaundy eliminant construction indeed provides annihilating curves for commuting elements in the q-deformed Heisenberg algebras for q not a root of unity. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
in
Generalized Lie Theory in Mathematics, Physics and Beyond
pages
16 pages
publisher
Springer
conference name
International Workshop of Baltic-Nordic Algebra, Geometry and Mathematical Physics
external identifiers
  • WOS:000264638600023
  • Scopus:58149162556
ISBN
978-3-540-85331-2
DOI
10.1007/978-3-540-85332-9_23
language
English
LU publication?
yes
id
0f9898d0-6985-4ef9-bce1-a07dbc69f0e7 (old id 2799095)
date added to LUP
2012-06-19 11:34:57
date last changed
2016-10-13 04:45:42
@misc{0f9898d0-6985-4ef9-bce1-a07dbc69f0e7,
  abstract     = {The aim of this paper to draw attention to several aspects of the algebraic dependence in algebras. The article starts with discussions of the algebraic dependence problem in commutative algebras. Then the Burchnall-Chaundy construction for proving algebraic dependence and obtaining the corresponding algebraic curves for commuting differential operators in the Heisenberg algebra is reviewed. Next some old and new results on algebraic dependence of commuting q-difference operators and elements in q-deformed Heisenberg algebras are reviewed. The main ideas and essence of two proofs of this are reviewed and compared. One is the algorithmic dimension growth existence proof. The other is the recent proof extending the Burchnall-Chaundy approach from differential operators and the Heisenberg algebra to the q-deformed Heisenberg algebra, showing that the Burchnall-Chaundy eliminant construction indeed provides annihilating curves for commuting elements in the q-deformed Heisenberg algebras for q not a root of unity.},
  author       = {Silvestrov, Sergei and Svensson, Charlotte and de Jeu, M.},
  isbn         = {978-3-540-85331-2},
  language     = {eng},
  pages        = {265--280},
  publisher    = {ARRAY(0x875f0d0)},
  series       = {Generalized Lie Theory in Mathematics, Physics and Beyond},
  title        = {Algebraic Dependence of Commuting Elements in Algebras},
  url          = {http://dx.doi.org/10.1007/978-3-540-85332-9_23},
  year         = {2009},
}