| Title | On algebraic curves for commuting elements in $q$-Heisenberg algebras |
| Authors | Johan Richter, Sergei Silvestrov |
| Alternative Location | http://www.ashdin.com/journ... |
| Publication | Journal of Generalized Lie Theory and Applications |
| Year | 2009 |
| Volume | 3 |
| Issue | 4 |
| Pages | 321 - 328 |
| Document type | Article |
| Status | Published |
| Quality controlled | Yes |
| Language | eng |
| Publisher | Ashdin Publishing |
| Abstract English | In the present article we continue investigating the algebraic dependence of commuting<br> elements in q-deformed Heisenberg algebras. We provide a simple proof that the<br> 0-chain subalgebra is a maximal commutative subalgebra when q is of free type and that<br> it coincides with the centralizer (commutant) of any one of its elements dierent from<br> the scalar multiples of the unity. We review the Burchnall-Chaundy-type construction for<br> proving algebraic dependence and obtaining corresponding algebraic curves for commuting<br> elements in the q-deformed Heisenberg algebra by computing a certain determinant<br> with entries depending on two commuting variables and one of the generators. The coe<br> cients in front of the powers of the generator in the expansion of the determinant are<br> polynomials in the two variables dening some algebraic curves and annihilating the two<br> commuting elements. We show that for the elements from the 0-chain subalgebra exactly<br> one algebraic curve arises in the expansion of the determinant. Finally, we present several<br> examples of computation of such algebraic curves and also make some observations on<br> the properties of these curves. |
| Keywords | Burchnall-Chaundy theory, Heisenberg algebra, |
| ISBN/ISSN/Other | ISSN: 1736-4337 (online) |
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