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Algebraic Dependence of Commuting Elements in Algebras

In: Generalized Lie Theory in Mathematics, Physics and Beyond

Author

Listed:
  • Sergei Silvestrov

    (Lund Institute of Technology, Lund University, Centre for Mathematical Sciences, Division of Mathematics)

  • Christian Svensson

    (Lund Institute of Technology, Lund University, Centre for Mathematical Sciences, Division of Mathematics)

  • Marcel de Jeu

    (Leiden University, Mathematical Institute)

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.

Suggested Citation

  • Sergei Silvestrov & Christian Svensson & Marcel de Jeu, 2009. "Algebraic Dependence of Commuting Elements in Algebras," Springer Books, in: Sergei Silvestrov & Eugen Paal & Viktor Abramov & Alexander Stolin (ed.), Generalized Lie Theory in Mathematics, Physics and Beyond, chapter 23, pages 265-280, Springer.
    • Handle: RePEc:spr:sprchp:978-3-540-85332-9_23
    DOI: 10.1007/978-3-540-85332-9_23
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