Journal of Generalized Lie Theory and Applications

On algebraic curves for commuting elements in q-Heisenberg algebras

Johan Richter and Sergei Silvestrov

Full-text: Open access

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 different 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 coefficients in front of the powers of the generator in the expansion of the determinant are polynomials in the two variables defining 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.

Article information

Source
J. Gen. Lie Theory Appl., Volume 3, Number 4 (2009), 321-328.

Dates
First available in Project Euclid: 6 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.jglta/1281106599

Digital Object Identifier
doi:10.4303/jglta/S090405

Mathematical Reviews number (MathSciNet)
MR2602994

Zentralblatt MATH identifier
1266.16039

Subjects
Primary: 16S99: None of the above, but in this section 81S05: Canonical quantization, commutation relations and statistics 39A13: Difference equations, scaling ($q$-differences) [See also 33Dxx]

Keywords
Associative Rings Associative Algebras Quantum Theory Canonical Quantization Commutation Relations Commutation Statistics Difference Equations Functional Equations Difference scaling q-differences

Citation

Richter, Johan; Silvestrov, Sergei. On algebraic curves for commuting elements in q-Heisenberg algebras. J. Gen. Lie Theory Appl. 3 (2009), no. 4, 321--328. doi:10.4303/jglta/S090405. https://projecteuclid.org/euclid.jglta/1281106599


Export citation

References

  • J. L. Burchnall and T. W. Chaundy. Commutative ordinary differential operators. Proc. Lond. Math. Soc. (2), 21 (1922), 420–440.
  • J. L. Burchnall and T. W. Chaundy. Commutative ordinary differential operators. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 118 (1928), 557–583.
  • J. L. Burchnall and T. W. Chaundy. Commutative ordinary differential operators. II. The Identity $P^n=Q^m$. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 134 (1932), 471–485.
  • L. Hellström and S. Silvestrov. Commuting Elements in $q$-Deformed Heisenberg Algebras. World Scientific Publishing, River Edge, NJ, 2000.
  • L. Hellström and S. Silvestrov. Ergodipotent maps and commutativity of elements in non-commutative rings and algebras with twisted intertwining. J. Algebra, 314 (2007), 17–41.
  • M. de Jeu, C. Svensson, and S. Silvestrov. Algebraic curves for commuting elements in the $q$-deformed Heisenberg algebra. J. Algebra, 321 (2009), 1239–1255.
  • I. Krichever. Integration of nonlinear equations by the methods of algebraic geometry (Russian). Funktsional. Anal. i Priložen, 11 (1977), 15–31.
  • I. Krichever. Methods of algebraic geometry in the theory of nonlinear equations (Russian). Uspekhi Mat. Nauk, 32 (1977), 183–208.
  • D. Larsson. Burchnall-Chaundy theory, Ore extensions and $\sigma$-differential operators. U.U.D.M. Report 2008:45, Department of Mathematics, Uppsala University, 2008.
  • D. Larsson and S. D. Silvestrov. Burchnall-Chaundy theory for $q$-difference operators and $q$-deformed Heisenberg algebras. J. Nonlinear Math. Phys., 10 (2003), Suppl. 2, 95–106.
  • D. Mumford. An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg deVries equation and related nonlinear equation. Proceedings of the International Symposium on Algebraic Geometry (Kyoto University, Kyoto, 1977), 1978, 115–153. }