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On algebraic curves for commuting elements in $q$-Heisenberg algebras

Richter, Johan LU and Silvestrov, Sergei LU (2009) In Journal of Generalized Lie Theory and Applications 3(4). p.321-328
Abstract
In the present article we continue investigating the algebraic dependence of commuting

elements in q-deformed Heisenberg algebras. We provide a simple proof that the

0-chain subalgebra is a maximal commutative subalgebra when q is of free type and that

it coincides with the centralizer (commutant) of any one of its elements dierent from

the scalar multiples of the unity. We review the Burchnall-Chaundy-type construction for

proving algebraic dependence and obtaining corresponding algebraic curves for commuting

elements in the q-deformed Heisenberg algebra by computing a certain determinant

with entries depending on two commuting variables and one of the generators. The... (More)
In the present article we continue investigating the algebraic dependence of commuting

elements in q-deformed Heisenberg algebras. We provide a simple proof that the

0-chain subalgebra is a maximal commutative subalgebra when q is of free type and that

it coincides with the centralizer (commutant) of any one of its elements dierent from

the scalar multiples of the unity. We review the Burchnall-Chaundy-type construction for

proving algebraic dependence and obtaining corresponding algebraic curves for commuting

elements in the q-deformed Heisenberg algebra by computing a certain determinant

with entries depending on two commuting variables and one of the generators. The coe

cients in front of the powers of the generator in the expansion of the determinant are

polynomials in the two variables dening some algebraic curves and annihilating the two

commuting elements. We show that for the elements from the 0-chain subalgebra exactly

one algebraic curve arises in the expansion of the determinant. Finally, we present several

examples of computation of such algebraic curves and also make some observations on

the properties of these curves. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Burchnall-Chaundy theory, Heisenberg algebra
in
Journal of Generalized Lie Theory and Applications
volume
3
issue
4
pages
321 - 328
publisher
Ashdin Publishing
ISSN
1736-5279
language
English
LU publication?
yes
id
a6b5aeb4-3b8d-4880-b845-daabfe365f85 (old id 2440747)
alternative location
http://www.ashdin.com/journals/jglta/2009/4/v3_n4_5.pdf
date added to LUP
2012-08-31 15:57:29
date last changed
2016-04-15 20:00:57
@article{a6b5aeb4-3b8d-4880-b845-daabfe365f85,
  abstract     = {In the present article we continue investigating the algebraic dependence of commuting<br/><br>
elements in q-deformed Heisenberg algebras. We provide a simple proof that the<br/><br>
0-chain subalgebra is a maximal commutative subalgebra when q is of free type and that<br/><br>
it coincides with the centralizer (commutant) of any one of its elements dierent from<br/><br>
the scalar multiples of the unity. We review the Burchnall-Chaundy-type construction for<br/><br>
proving algebraic dependence and obtaining corresponding algebraic curves for commuting<br/><br>
elements in the q-deformed Heisenberg algebra by computing a certain determinant<br/><br>
with entries depending on two commuting variables and one of the generators. The coe<br/><br>
cients in front of the powers of the generator in the expansion of the determinant are<br/><br>
polynomials in the two variables dening some algebraic curves and annihilating the two<br/><br>
commuting elements. We show that for the elements from the 0-chain subalgebra exactly<br/><br>
one algebraic curve arises in the expansion of the determinant. Finally, we present several<br/><br>
examples of computation of such algebraic curves and also make some observations on<br/><br>
the properties of these curves.},
  author       = {Richter, Johan and Silvestrov, Sergei},
  issn         = {1736-5279},
  keyword      = {Burchnall-Chaundy theory,Heisenberg algebra},
  language     = {eng},
  number       = {4},
  pages        = {321--328},
  publisher    = {Ashdin Publishing},
  series       = {Journal of Generalized Lie Theory and Applications},
  title        = {On algebraic curves for commuting elements in $q$-Heisenberg algebras},
  volume       = {3},
  year         = {2009},
}